christopher/oeis · Datasets at Hugging Face

sequence_id stringlengths 7 7	sequence_name stringlengths 4 573	sequence listlengths 1 348	keywords listlengths 1 7	score int64 1 2.47k	offset_a int64 -14,827 666,262,453B	offset_b int64 0 635M ⌀	cross_references listlengths 1 128 ⌀	former_ids listlengths 1 3 ⌀	author stringlengths 7 231 ⌀	timestamp timestamp[us]date 1999-12-11 03:00:00 2026-01-19 02:46:49	filename stringlengths 29 29	hash stringlengths 32 32
A392000	Array read by antidiagonals: A(k, n) is the number of undirected closed Hamiltonian paths on the k X n knight graph.	[ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "...	[ "nonn", "tabl" ]	15	1	50	[ "A000004", "A001230", "A169764", "A175855", "A175881", "A193054", "A193055", "A390833", "A391999", "A392000" ]	null	Peter Luschny, Dec 26 2025	2025-12-27T13:09:56	oeisdata/seq/A392/A392000.seq	6e9b9973be577742bc33e83c3672430d
A392001	Numbers k such that k + 1 is not a power of an odd prime and 2*k + 1 is not a prime power.	[ "7", "17", "19", "25", "27", "31", "32", "34", "37", "38", "43", "45", "47", "49", "55", "57", "59", "61", "64", "67", "71", "73", "76", "77", "79", "85", "87", "91", "92", "93", "94", "97", "101", "103", "104", "107", "109", "110", "115", ...	[ "nonn", "changed" ]	25	1	1	[ "A003418", "A045979", "A053176", "A067656", "A093880", "A166602", "A246655", "A392001" ]	null	Peter Luschny, Dec 31 2025	2026-01-17T20:15:06	oeisdata/seq/A392/A392001.seq	6b6e535814cd96b3c06595971d76a4b7
A392002	Number of partitions of n with parts colored by {0, 1} such that the sum of colors is congruent to 1 (mod 2).	[ "0", "1", "2", "5", "9", "18", "31", "55", "90", "150", "237", "376", "577", "885", "1325", "1978", "2900", "4235", "6100", "8745", "12400", "17501", "24477", "34075", "47079", "64756", "88493", "120420", "162940", "219595", "294476", "393407", "523237...	[ "nonn" ]	11	0	3	[ "A000041", "A000712", "A387957", "A390208", "A392002" ]	null	Thomas Hutton, Dec 26 2025	2026-01-01T17:33:09	oeisdata/seq/A392/A392002.seq	db28f4c59031632aa50add244f6f47e6
A392003	a(n) = (72^n - 9 + 5(-1)^n) / 6.	[ "0", "4", "7", "18", "35", "74", "147", "298", "595", "1194", "2387", "4778", "9555", "19114", "38227", "76458", "152915", "305834", "611667", "1223338", "2446675", "4893354", "9786707", "19573418", "39146835", "78293674", "156587347", "313174698", "626349395"...	[ "nonn", "easy" ]	30	1	2	[ "A266753", "A390768", "A392003" ]	null	Karl-Heinz Hofmann, Dec 26 2025	2026-01-02T08:29:06	oeisdata/seq/A392/A392003.seq	b6f8d82f63e5c44fca18d9f2ba373369
A392004	Numbers k such that the base-k digits of 2^k include at least one 0.	[ "2", "4", "8", "10", "16", "19", "31", "32", "35", "39", "46", "48", "61", "64", "71", "93", "102", "103", "104", "108", "109", "117", "122", "128", "139", "154", "158", "166", "171", "181", "184", "185", "192", "193", "199", "218", "226", "2...	[ "nonn", "base" ]	18	1	1	[ "A068535", "A392004" ]	null	Robert Israel, Dec 26 2025	2025-12-30T11:41:50	oeisdata/seq/A392/A392004.seq	10e136873c24ae7b8acdb8f6a078bb02
A392007	Smallest number k such that log_b(k) - floor(log_b(k)) >= 1/2 for b = 2..n, and not for b = n+1.	[ "3", "6", "47", "56", "920483", "46341", "888667668", "13207922282067" ]	[ "nonn", "more", "new" ]	50	2	1	[ "A002163", "A002193", "A002194", "A010464", "A020857", "A153459", "A392007" ]	null	Jakub Buczak, Dec 26 2025	2026-01-05T14:05:15	oeisdata/seq/A392/A392007.seq	fe9019a5879694cd0616a242824c2083
A392008	Odd squarefree integers with at least 3 prime factors.	[ "105", "165", "195", "231", "255", "273", "285", "345", "357", "385", "399", "429", "435", "455", "465", "483", "555", "561", "595", "609", "615", "627", "645", "651", "663", "665", "705", "715", "741", "759", "777", "795", "805", "861", "885", "...	[ "nonn", "changed" ]	45	1	1	[ "A001221", "A001222", "A046389", "A350352", "A392008", "A392009" ]	null	Charles Kusniec, Dec 26 2025	2026-01-08T10:10:41	oeisdata/seq/A392/A392008.seq	bd1123b73f1431610754232c118d53ea
A392009	Even squarefree integers with at least 3 prime factors.	[ "30", "42", "66", "70", "78", "102", "110", "114", "130", "138", "154", "170", "174", "182", "186", "190", "210", "222", "230", "238", "246", "258", "266", "282", "286", "290", "310", "318", "322", "330", "354", "366", "370", "374", "390", "402",...	[ "nonn", "changed" ]	36	1	1	[ "A001221", "A001222", "A053858", "A075819", "A350352", "A392008", "A392009" ]	null	Charles Kusniec, Dec 26 2025	2026-01-08T09:40:57	oeisdata/seq/A392/A392009.seq	134d74c1ae92b07ff47cfd3a101f9963
A392012	The sum of the exponential divisors of the cubefree numbers.	[ "1", "2", "3", "6", "5", "6", "7", "12", "10", "11", "18", "13", "14", "15", "17", "24", "19", "30", "21", "22", "23", "30", "26", "42", "29", "30", "31", "33", "34", "35", "72", "37", "38", "39", "41", "42", "43", "66", "60", "46", "47...	[ "nonn", "easy" ]	11	1	2	[ "A002117", "A004709", "A013662", "A051377", "A057723", "A322791", "A322857", "A382419", "A392012" ]	null	Amiram Eldar, Dec 27 2025	2025-12-28T09:57:15	oeisdata/seq/A392/A392012.seq	15b5c7eabc32b6e02f29eb138023a83e
A392013	The number of exponential divisors of the exponentially-2^n numbers.	[ "1", "1", "1", "2", "1", "1", "1", "2", "1", "1", "2", "1", "1", "1", "3", "1", "2", "1", "2", "1", "1", "1", "2", "1", "2", "1", "1", "1", "1", "1", "1", "4", "1", "1", "1", "1", "1", "1", "2", "2", "1", "1", "3", "2", "2", "...	[ "nonn", "easy" ]	8	1	4	[ "A049419", "A138302", "A271727", "A322791", "A392013", "A392014", "A392015" ]	null	Amiram Eldar, Dec 27 2025	2025-12-28T09:57:23	oeisdata/seq/A392/A392013.seq	ce4813de9e394438741ec9178f03b6d9
A392014	The number of exponential unitary divisors of the exponentially-2^n numbers.	[ "1", "1", "1", "2", "1", "1", "1", "2", "1", "1", "2", "1", "1", "1", "2", "1", "2", "1", "2", "1", "1", "1", "2", "1", "2", "1", "1", "1", "1", "1", "1", "4", "1", "1", "1", "1", "1", "1", "2", "2", "1", "1", "2", "2", "2", "...	[ "nonn", "easy" ]	8	1	4	[ "A138302", "A278908", "A361255", "A366538", "A392013", "A392014", "A392016" ]	null	Amiram Eldar, Dec 27 2025	2025-12-28T09:57:18	oeisdata/seq/A392/A392014.seq	3000ca0ddfc68ba54c3e279cbcd45f9c
A392015	The sum of the exponential divisors of the exponentially-2^n numbers.	[ "1", "1", "1", "3", "1", "1", "1", "4", "1", "1", "3", "1", "1", "1", "7", "1", "4", "1", "3", "1", "1", "1", "6", "1", "3", "1", "1", "1", "1", "1", "1", "12", "1", "1", "1", "1", "1", "1", "3", "4", "1", "1", "7", "8", "6", ...	[ "nonn", "easy" ]	8	1	4	[ "A051377", "A138302", "A322791", "A392013", "A392015", "A392016" ]	null	Amiram Eldar, Dec 27 2025	2025-12-28T09:57:27	oeisdata/seq/A392/A392015.seq	dd315e2e827ab5bfe18ad24120d01fdd
A392016	The sum of the exponential unitary divisors of the exponentially-2^n numbers.	[ "1", "2", "3", "6", "5", "6", "7", "12", "10", "11", "18", "13", "14", "15", "18", "17", "24", "19", "30", "21", "22", "23", "30", "26", "42", "29", "30", "31", "33", "34", "35", "72", "37", "38", "39", "41", "42", "43", "66", "60", "46...	[ "nonn", "easy" ]	8	1	2	[ "A138302", "A271727", "A322857", "A361255", "A392014", "A392015", "A392016" ]	null	Amiram Eldar, Dec 27 2025	2025-12-28T09:57:31	oeisdata/seq/A392/A392016.seq	139c433d8c7361fcd7732f53a2237110
A392017	Decimal expansion of sqrt(3)*log(2+sqrt(3)).	[ "2", "2", "8", "1", "0", "3", "7", "9", "8", "8", "9", "0", "2", "8", "3", "9", "0", "4", "2", "5", "9", "3", "2", "8", "2", "7", "6", "4", "6", "4", "1", "2", "1", "7", "4", "7", "2", "6", "0", "4", "8", "8", "3", "1", "9", "...	[ "nonn", "cons", "easy" ]	24	1	1	[ "A065918", "A196530", "A257436", "A259830", "A392017" ]	null	Artur Jasinski, Dec 27 2025	2026-01-01T17:57:37	oeisdata/seq/A392/A392017.seq	70eb036ddf9d1020fb77dcbae2b22aeb
A392019	If binomial(n,k) = u*v where the only primes dividing u are <= k and the only primes dividing v are > k, then a(n) is the least k such that u > n^2, or 0 if such a k doesn't exist.	[ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "8", "0", "0", "0", "12", "13", "13", "5", "0", "7", "0", "9", "12", "11", "11", "7", "11", "17", "15", "14", "17", "17", "19", "1...	[ "nonn" ]	12	1	21	[ "A007318", "A392019" ]	null	Elijah Beregovsky, Dec 27 2025	2026-01-01T22:16:27	oeisdata/seq/A392/A392019.seq	50439b8533cc1193d3e8768986cad290
A392020	Expansion of the series e(-q), where the series e(q) is Morier-Genoud and Ovsienko's q-analog of Euler's number e.	[ "1", "-1", "0", "-1", "0", "1", "2", "3", "3", "1", "-3", "-9", "-17", "-25", "-29", "-23", "2", "54", "134", "232", "320", "347", "243", "-71", "-660", "-1531", "-2575", "-3504", "-3804", "-2747", "488", "6537", "15395", "25819", "34716", "36780...	[ "sign", "easy", "new" ]	28	0	7	[ "A001113", "A003417", "A004148", "A337589", "A392020", "A392021", "A392022", "A392023" ]	null	Peter Bala, Dec 27 2025	2026-01-11T09:20:57	oeisdata/seq/A392/A392020.seq	9c7f73d23b203212143951a52f5d3385
A392021	Expansion of the series 1/e(-q), where the series e(q) is Morier-Genoud and Ovsienko's q-analog of the real number e = exp(1).	[ "1", "1", "1", "2", "3", "3", "2", "-1", "-8", "-20", "-37", "-57", "-71", "-59", "12", "190", "531", "1073", "1785", "2476", "2657", "1381", "-2877", "-12163", "-28550", "-52828", "-81924", "-104671", "-96015", "-10924", "218528", "675902", "1431270",...	[ "sign", "easy", "new" ]	18	0	4	[ "A001113", "A003417", "A337589", "A392020", "A392021", "A392022", "A392023" ]	null	Peter Bala, Jan 04 2026	2026-01-11T09:21:01	oeisdata/seq/A392/A392021.seq	c45d191c4145e4f7b65a60f983a03b99
A392022	Expansion of the series e(q)*e(-q) (even coefficients only), where the series e(q) is Morier-Genoud and Ovsienko's q-analog of the real number e = exp(1).	[ "1", "-1", "-2", "5", "14", "1", "-52", "-125", "-83", "347", "1217", "1579", "-949", "-8529", "-17399", "-10159", "40670", "134560", "167114", "-104686", "-858254", "-1591711", "-453531", "4997043", "13433932", "12368077", "-22722489", "-102188424", "-1567129...	[ "sign", "easy", "new" ]	17	0	3	[ "A001113", "A003417", "A337589", "A392020", "A392021", "A392022", "A392023" ]	null	Peter Bala, Jan 05 2026	2026-01-11T09:21:06	oeisdata/seq/A392/A392022.seq	6ecb30bb0cab3799ae9f1d10009aa54e
A392023	Expansion of the series e(q)/e(-q), where the series e(q) is Morier-Genoud and Ovsienko's q-analog of the real number e = exp(1).	[ "1", "2", "2", "4", "6", "6", "8", "2", "-6", "-26", "-62", "-96", "-156", "-150", "-106", "150", "698", "1590", "3142", "4630", "6234", "5436", "528", "-12876", "-40696", "-83918", "-148010", "-210998", "-248122", "-175238", "126894", "823188", "21380...	[ "sign", "easy", "new" ]	13	0	2	[ "A001113", "A003417", "A337589", "A392020", "A392021", "A392022", "A392023" ]	null	Peter Bala, Jan 06 2026	2026-01-11T09:21:10	oeisdata/seq/A392/A392023.seq	dc71d90f5d45f5123dac00eb0b64a89b
A392025	Least m such that n can be represented as a linear combination of 1^3, 2^3, ..., m^3 with coefficients +-1 only.	[ "12", "1", "12", "13", "8", "9", "8", "2", "15", "2", "15", "9", "11", "14", "11", "10", "11", "10", "3", "10", "3", "13", "8", "13", "16", "5", "7", "5", "4", "14", "4", "13", "7", "10", "3", "10", "3", "13", "11", "10", "11", "5", ...	[ "nonn", "look" ]	23	0	1	[ "A231015", "A392025" ]	null	Jeffrey Shallit, Dec 27 2025	2025-12-31T22:20:54	oeisdata/seq/A392/A392025.seq	bcc95d620ef0cec77d12f923e21dc7dc
A392026	Decimal expansion of 113*Pi + 355.	[ "7", "0", "9", "9", "9", "9", "9", "6", "9", "8", "5", "5", "6", "4", "6", "6", "3", "5", "9", "4", "6", "2", "7", "8", "7", "0", "2", "3", "1", "0", "5", "8", "3", "8", "2", "5", "9", "1", "4", "2", "8", "0", "1", "4", "2", "...	[ "nonn", "cons" ]	26	3	1	[ "A014307", "A068079", "A180875", "A226043", "A392026" ]	null	Artur Jasinski, Dec 27 2025	2026-01-02T13:52:24	oeisdata/seq/A392/A392026.seq	976ece7e1c2a789f2eeaefeff38cc0b5
A392027	Partial sums of A255933.	[ "1", "3", "5", "8", "10", "13", "15", "18", "20", "23", "27", "29", "31", "33", "35", "38", "40", "43", "45", "48", "50", "53", "55", "58", "60", "63", "65", "68", "70", "73", "75", "78", "80", "83", "85", "88", "90", "93", "95", "98", ...	[ "nonn" ]	11	1	2	[ "A255933", "A392027", "A392028" ]	null	Paolo Xausa, Dec 27 2025	2025-12-28T09:54:48	oeisdata/seq/A392/A392027.seq	d888708f6a40512cdbedff7d5086865c
A392028	Indices of records in A255933.	[ "1", "2", "4", "11", "49", "286", "1997", "31925", "287310", "1436559", "31604222", "189625346", "4930259198", "34511814535" ]	[ "nonn", "hard", "more" ]	14	1	2	[ "A255933", "A392027", "A392028" ]	null	Paolo Xausa, Dec 27 2025	2025-12-29T12:25:58	oeisdata/seq/A392/A392028.seq	108ee240f8e4509bec605fccc43505fa
A392030	Numbers k with squarefree kernel 210 such that both k-1 and k+1 are prime.	[ "420", "1050", "3360", "5880", "6300", "7350", "7560", "8820", "10500", "26250", "26880", "28350", "29400", "30870", "33600", "35280", "100800", "102060", "131250", "134400", "158760", "170100", "241920", "264600", "282240", "294000", "370440", "420000", "4536...	[ "nonn", "easy", "new" ]	6	1	1	[ "A002110", "A007947", "A014574", "A027856", "A147571", "A386252", "A392030" ]	null	Michael De Vlieger, Jan 06 2026	2026-01-08T09:28:01	oeisdata/seq/A392/A392030.seq	0457871d6219aa5537c5b66f382bed2d
A392031	a(n) is the smallest k which has exactly n divisors d such that sigma(d) = sigma(d + k/d) where sigma = A000203.	[ "1", "14", "120", "4080", "132132", "7361280" ]	[ "nonn", "more" ]	12	0	2	[ "A000005", "A000203", "A392031" ]	null	Juri-Stepan Gerasimov, Dec 27 2025	2026-01-04T12:14:13	oeisdata/seq/A392/A392031.seq	8007a1e3e024f952ee5164e7cd696f05
A392032	a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(n,k) * binomial(n-k,k).	[ "1", "1", "7", "19", "73", "241", "831", "2787", "9339", "31003", "102373", "336073", "1097911", "3570607", "11565529", "37324437", "120052107", "384962859", "1230978093", "3926097681", "12492150819", "39660238779", "125656218933", "397360559841", "1254333690477", "...	[ "nonn" ]	14	0	3	[ "A002426", "A098473", "A391990", "A392032", "A392033" ]	null	Seiichi Manyama, Dec 27 2025	2025-12-31T14:26:13	oeisdata/seq/A392/A392032.seq	d51229e9b48b86f850024859b68f09af
A392033	a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(n,k) * binomial(n-k,k).	[ "1", "1", "9", "25", "109", "381", "1421", "5069", "18075", "63499", "221323", "764523", "2621587", "8928115", "30220347", "101721707", "340659227", "1135543323", "3769023795", "12460643043", "41045661279", "134748546639", "440974812831", "1438893866655", "46822262130...	[ "nonn" ]	15	0	3	[ "A002426", "A098473", "A391990", "A392032", "A392033" ]	null	Seiichi Manyama, Dec 27 2025	2025-12-31T12:34:33	oeisdata/seq/A392/A392033.seq	c622caefa74a66a3436986aada26f278
A392034	a(n) = prime(n)*(prime(n+1) - prime(n))/2.	[ "1", "3", "5", "14", "11", "26", "17", "38", "69", "29", "93", "74", "41", "86", "141", "159", "59", "183", "134", "71", "219", "158", "249", "356", "194", "101", "206", "107", "218", "791", "254", "393", "137", "695", "149", "453", "471", "3...	[ "nonn", "look", "new" ]	26	1	2	[ "A000040", "A001223", "A291463", "A392034" ]	null	Anthony S. Wright, Dec 27 2025	2026-01-10T20:26:38	oeisdata/seq/A392/A392034.seq	f90f1ca0700f401a176e73e2dc3736b0
A392035	Irregular table read by rows: the n-th row contains, in weakly decreasing order, the positive integers whose squares sum to n as obtained by the greedy algorithm.	[ "1", "1", "1", "1", "1", "1", "2", "2", "1", "2", "1", "1", "2", "1", "1", "1", "2", "2", "3", "3", "1", "3", "1", "1", "3", "1", "1", "1", "3", "2", "3", "2", "1", "3", "2", "1", "1", "4", "4", "1", "4", "1", "1", "4", "1", "...	[ "nonn", "tabf" ]	10	1	7	[ "A000196", "A053610", "A350674", "A392035" ]	null	Stefano Spezia, Dec 28 2025	2025-12-29T07:33:59	oeisdata/seq/A392/A392035.seq	d37e20c69416e913cdbcbce9c83272d1
A392036	Irregular table read by rows: the n-th row contains, in weakly decreasing order, the powers of 2 summing to n as obtained by the greedy algorithm.	[ "1", "2", "2", "1", "4", "4", "1", "4", "2", "4", "2", "1", "8", "8", "1", "8", "2", "8", "2", "1", "8", "4", "8", "4", "1", "8", "4", "2", "8", "4", "2", "1", "16", "16", "1", "16", "2", "16", "2", "1", "16", "4", "16", "4", "1...	[ "nonn", "tabf" ]	12	1	2	[ "A000079", "A000120", "A006519", "A053644", "A272011", "A392036" ]	null	Stefano Spezia, Dec 28 2025	2026-01-02T09:31:16	oeisdata/seq/A392/A392036.seq	db152c379c30cbe616b8c21bb77a2dfe
A392037	Square table read by downward antidiagonals: n-th row has e.g.f. (1-9*x)^(-n/9).	[ "1", "0", "1", "0", "1", "1", "0", "10", "2", "1", "0", "190", "22", "3", "1", "0", "5320", "440", "36", "4", "1", "0", "196840", "12760", "756", "52", "5", "1", "0", "9054640", "484880", "22680", "1144", "70", "6", "1", "0", "498005200", "...	[ "nonn", "easy", "tabl" ]	9	0	8	[ "A035012", "A035013", "A035017", "A035018", "A035020", "A035021", "A035022", "A035023", "A045756", "A049211", "A051232", "A053116", "A084949", "A144758", "A144829", "A147629", "A147630", "A147631", "A392037" ]	null	Jianing Song, Dec 29 2025	2025-12-29T07:31:21	oeisdata/seq/A392/A392037.seq	614f6b0cc0553c469aec24c970803712
A392038	Least number k such that n can be represented as a linear combination of 1^n, 2^n, ..., k^n with coefficients +-1 only.	[ "2", "1", "4", "13", "15", "25", "31", "25", "47", "57", "60" ]	[ "nonn", "more", "new" ]	59	0	1	[ "A019568", "A126908", "A231015", "A392025", "A392038" ]	null	Jean-Marc Rebert, Dec 29 2025	2026-01-17T17:23:15	oeisdata/seq/A392/A392038.seq	a30159461b9e51b5ab77021df019f227
A392041	Smallest prime number whose decimal digits include the consecutive pattern 1, 2, ..., n.	[ "11", "127", "1123", "12343", "123457", "12345623", "71234567", "1234567801", "1234567891", "1234567891003", "12345678910111", "12345678910111207", "912345678910111213", "123456789101112131449", "12345678910111213141523", "123456789101112131415161", "41234567891011121314151617", "3...	[ "nonn", "base" ]	27	1	1	[ "A007908", "A053546", "A058183", "A176942", "A392041" ]	null	Jean-Marc Rebert, Dec 29 2025	2026-01-04T13:32:47	oeisdata/seq/A392/A392041.seq	ab5361b59d64affdae698b4eb12b4e37
A392042	a(n) = Sum_{k=0..floor(n/3)} 2^k * 3^(n-3k) binomial(k,3(n-3k)).	[ "1", "0", "0", "2", "0", "0", "4", "0", "0", "8", "24", "0", "16", "192", "0", "32", "960", "0", "64", "3840", "576", "128", "13440", "8064", "256", "43008", "64512", "512", "129024", "387072", "14848", "368640", "1935360", "278528", "1013760", "...	[ "nonn" ]	18	0	4	[ "A178618", "A392042", "A392043" ]	null	Seiichi Manyama, Dec 29 2025	2025-12-31T08:49:15	oeisdata/seq/A392/A392042.seq	fb45a442cfe2483df0d3d40f7a6583e7
A392043	a(n) = Sum_{k=0..floor(n/3)} (k+1) * 2^k * 3^(n-3k) binomial(k,3(n-3k)).	[ "1", "0", "0", "4", "0", "0", "12", "0", "0", "32", "96", "0", "80", "960", "0", "192", "5760", "0", "448", "26880", "4032", "1024", "107520", "64512", "2304", "387072", "580608", "5120", "1290240", "3870720", "149504", "4055040", "21288960", "306585...	[ "nonn" ]	18	0	4	[ "A178618", "A392042", "A392043" ]	null	Seiichi Manyama, Dec 29 2025	2025-12-31T08:49:10	oeisdata/seq/A392/A392043.seq	9cedcf924a08bc8f5b2dc16b9e84521b
A392044	a(n) = Sum_{k=0..n} (k+1) * binomial(k,3*(n-k)).	[ "1", "2", "3", "4", "9", "26", "67", "148", "296", "570", "1103", "2172", "4313", "8528", "16691", "32360", "62365", "119830", "229840", "440052", "840653", "1602038", "3045967", "5779676", "10947997", "20706438", "39107943", "73764072", "138955192", "261449...	[ "nonn" ]	19	0	2	[ "A001629", "A003522", "A178618", "A391962", "A392044" ]	null	Seiichi Manyama, Dec 29 2025	2025-12-31T12:34:42	oeisdata/seq/A392/A392044.seq	da9cdce0caf9d12e8a5409e65a2d9106
A392045	Odd positive integers whose canonical prime factorization p_1^e_1...p_r^e_r satisfies that {p_1, ..., p_r, e_1, ..., e_r} is a set of 2*r consecutive integers.	[ "9", "81", "625", "2025", "5625", "15625", "117649", "455625", "1265625", "5764801", "22325625", "37515625", "43758225", "62015625", "73530625", "238239225", "337640625", "661775625", "25937424601", "45956640625", "90075015625", "246140015625", "482434430625", "68372226...	[ "nonn", "easy", "new" ]	8	1	1	[ "A097320", "A111774", "A382330", "A383397", "A387172", "A387586", "A389339", "A389340", "A389795", "A390784", "A392045", "A392046", "A392047" ]	null	Felix Huber, Jan 05 2026	2026-01-08T23:14:53	oeisdata/seq/A392/A392045.seq	f04bf9b0f8db9dc0e5824e83f4f484e6
A392046	Even positive integers whose canonical prime factorization p_1^e_1...p_r^e_r satisfies that {p_1, ..., p_r, e_1, ..., e_r} is a set of 2*r consecutive integers.	[ "2", "8", "48", "162", "2000", "2592", "3888", "5000", "25920", "58320", "120000", "750000", "911250", "2531250", "13720000", "19208000", "37340352", "56010528", "58320000", "85750000", "87127488", "87480000", "162000000", "196036848", "235298000", "300125000", "3...	[ "nonn", "easy", "fini", "full", "new" ]	8	1	1	[ "A097320", "A111774", "A382330", "A383397", "A387172", "A387586", "A389339", "A389340", "A389795", "A390784", "A392045", "A392046", "A392047" ]	null	Felix Huber, Jan 05 2026	2026-01-09T16:08:13	oeisdata/seq/A392/A392046.seq	1d3ba6c371482a1cf937e0ded265e87c
A392047	Positive integers whose canonical prime factorization p_1^e_1...p_r^e_r satisfies that {p_1, ..., p_r, e_1, ..., e_r} is a set of 2*r consecutive integers.	[ "2", "8", "9", "48", "81", "162", "625", "2000", "2025", "2592", "3888", "5000", "5625", "15625", "25920", "58320", "117649", "120000", "455625", "750000", "911250", "1265625", "2531250", "5764801", "13720000", "19208000", "22325625", "37340352", "37515625", ...	[ "nonn", "easy", "new" ]	12	1	1	[ "A097320", "A111774", "A382330", "A383397", "A387172", "A387586", "A389339", "A389340", "A389795", "A390784", "A392045", "A392046", "A392047" ]	null	Felix Huber, Jan 05 2026	2026-01-09T16:13:09	oeisdata/seq/A392/A392047.seq	e1330bf8981a16a047798fbafd61ed81
A392048	a(n) is the number of distinct integers that can be created by summing combinations of the n+1 first rationals {b(0)/b(1), ..., b(n)/b(n+1)}, where b(k)=A002487(k) is Stern's diatomic series.	[ "1", "2", "2", "4", "4", "6", "7", "10", "10", "11", "11", "13", "14", "16", "17", "21", "21", "22", "22", "24", "24", "25", "25", "29", "30", "32", "33", "36", "37", "39", "40", "45", "45", "46", "46", "48", "48", "49", "49", "53", "53...	[ "nonn", "new" ]	11	0	2	[ "A002487", "A392048" ]	null	Frederik G. Faye, Dec 29 2025	2026-01-04T22:32:51	oeisdata/seq/A392/A392048.seq	7c79a22ef47c8805d8b2c0d9c7b91e1a
A392049	a(n) is the numerator of the harmonic mean of the digits of n.	[ "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "0", "1", "4", "3", "8", "5", "12", "7", "16", "9", "0", "4", "2", "12", "8", "20", "3", "28", "16", "36", "0", "3", "12", "3", "24", "15", "4", "21", "48", "9", "0", "8", "8", "24...	[ "nonn", "base", "easy", "frac" ]	11	0	3	[ "A011540", "A371383", "A392049", "A392050", "A392051" ]	null	Stefano Spezia, Dec 29 2025	2026-01-02T09:28:07	oeisdata/seq/A392/A392049.seq	fb37ae93deefd359387f173f620db166
A392050	a(n) is the denominator of the harmonic mean of the digits of n.	[ "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "3", "2", "5", "3", "7", "4", "9", "5", "1", "3", "1", "5", "3", "7", "1", "9", "5", "11", "1", "2", "5", "1", "7", "4", "1", "5", "11", "2", "1", "5", "3", "7", "1", ...	[ "nonn", "base", "easy", "frac" ]	8	0	13	[ "A011540", "A371384", "A392049", "A392050", "A392051" ]	null	Stefano Spezia, Dec 29 2025	2026-01-02T09:30:43	oeisdata/seq/A392/A392050.seq	0df9db7e1af64d222d4eb808bb562878
A392051	Nonnegative integers such that the harmonic mean of the digits is an integer.	[ "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "20", "22", "26", "30", "33", "36", "40", "44", "50", "55", "60", "62", "63", "66", "70", "77", "80", "88", "90", "99", "100", "101", "102", "103", "104", "105", "106", "107", ...	[ "nonn", "base", "easy" ]	8	1	3	[ "A011540", "A061383", "A061430", "A147591", "A392049", "A392050", "A392051" ]	null	Stefano Spezia, Dec 29 2025	2026-01-02T09:29:36	oeisdata/seq/A392/A392051.seq	e1e0ccb342569bbbe41ff69c1b3ec505
A392055	a(n) = p if n = p^k is a perfect power of either a ramified or inert prime in the Gaussian integers and 1 otherwise.	[ "1", "2", "3", "2", "1", "1", "7", "2", "3", "1", "11", "1", "1", "1", "1", "2", "1", "1", "19", "1", "1", "1", "23", "1", "1", "1", "3", "1", "1", "1", "31", "2", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "43", "1", "1"...	[ "nonn", "easy", "new" ]	28	1	2	[ "A002145", "A014963", "A045326", "A392055", "A392059" ]	null	Rick Zhou, Dec 29 2025	2026-01-05T16:21:34	oeisdata/seq/A392/A392055.seq	e76cae950dfebfa533c3840626cf4649
A392056	Numbers k such that (47^k - 4^k)/43 is prime.	[ "7", "12703", "15679", "49711", "90749" ]	[ "nonn", "hard", "more" ]	5	1	1	[ "A062587", "A062589", "A127996", "A127997", "A128344", "A204940", "A217320", "A225807", "A229542", "A375161", "A375236", "A377031", "A392056" ]	null	Robert Price, Dec 29 2025	2025-12-30T13:14:14	oeisdata/seq/A392/A392056.seq	4133511f4c6160678a3727ccd5d72b03
A392057	a(0)=0, a(n) = wt(a(n-1)) + wt(n), where wt(x) is the binary weight of x, A000120.	[ "0", "1", "2", "3", "3", "4", "3", "5", "3", "4", "3", "5", "4", "4", "4", "5", "3", "4", "3", "5", "4", "4", "4", "5", "4", "4", "4", "5", "5", "6", "6", "7", "4", "3", "4", "4", "3", "5", "5", "6", "4", "4", "4", "5", "5", "...	[ "nonn", "base", "easy", "new" ]	17	0	3	[ "A000120", "A000788", "A392057" ]	null	Alex Ratushnyak, Dec 29 2025	2026-01-10T20:30:05	oeisdata/seq/A392/A392057.seq	f32e1f38a8bbeb76cf2db3a867c60ebb
A392058	Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + Sum_{j=0..k} binomial(k+1,j)A(n-1,j)A(k-j,0) with A(0,k) = 1.	[ "1", "1", "2", "1", "5", "7", "1", "17", "31", "38", "1", "83", "178", "254", "292", "1", "578", "1330", "2099", "2683", "2975", "1", "5474", "12748", "21338", "29425", "35375", "38350", "1", "66932", "153136", "263870", "382966", "489383", "566083...	[ "nonn", "tabl" ]	6	0	3	[ "A233335", "A392058" ]	null	Mikhail Kurkov, Dec 29 2025	2026-01-04T20:28:11	oeisdata/seq/A392/A392058.seq	50abcfbcb5ec38ac470e802fc4d753d0
A392059	a(n) is the product of all p^k such that p is a ramified or inert prime in the Gaussian integers and k is the largest such k satisfying p^k <= n.	[ "1", "2", "6", "12", "12", "12", "84", "168", "504", "504", "5544", "5544", "5544", "5544", "5544", "11088", "11088", "11088", "210672", "210672", "210672", "210672", "4845456", "4845456", "4845456", "4845456", "14536368", "14536368", "14536368", "14536368",...	[ "nonn", "look", "easy", "new" ]	17	1	2	[ "A387569", "A392055", "A392059" ]	null	Rick Zhou, Dec 29 2025	2026-01-10T23:01:37	oeisdata/seq/A392/A392059.seq	a857d99b3e377a04e6b5c2f973766a5f
A392061	Irregular triangular array read by rows: T(n,k) is the number of periodic functions on [n] that have order k, n>=0, 1<=k<=A000793(n).	[ "1", "1", "3", "1", "10", "9", "2", "41", "69", "32", "6", "196", "555", "360", "150", "24", "20", "1057", "4845", "3680", "2520", "864", "840", "6322", "45843", "37800", "37590", "19152", "22050", "720", "0", "0", "504", "0", "420", "41393", "...	[ "nonn", "tabf", "look", "changed" ]	30	0	3	[ "A000248", "A000793", "A006153", "A057731", "A162682", "A199673", "A222029", "A392061" ]	null	Geoffrey Critzer, Dec 29 2025	2026-01-05T15:10:42	oeisdata/seq/A392/A392061.seq	7334f3ddc714a7d6e56eeab265c5e7d1
A392062	Numbers k such that every prime factor of the sum of decimal digits of k is greater than every prime factor of k.	[ "1", "10", "16", "25", "32", "49", "56", "98", "100", "128", "160", "175", "245", "250", "256", "289", "490", "539", "560", "595", "616", "625", "676", "728", "748", "784", "845", "847", "896", "968", "980", "1000", "1024", "1280", "1372", "1408"...	[ "nonn", "base" ]	10	1	2	[ "A006530", "A007953", "A011557", "A020639", "A392062" ]	null	Robert Israel, Dec 29 2025	2026-01-04T20:33:28	oeisdata/seq/A392/A392062.seq	5203faa7962e8427d9fef000db806016
A392064	Decimal expansion of the expected number of consecutive random decimal digits whose running sum first exceeds 9.	[ "2", "8", "6", "7", "9", "7", "1", "9", "9", "0", "7", "9", "2", "4", "4", "1", "3", "1", "3", "3", "2", "2", "2", "5", "7", "2", "3", "1", "2", "4", "0", "8", "3", "6", "9", "0", "6", "5", "6", "6", "1", "6", "5", "4", "0", "...	[ "nonn", "cons", "easy" ]	44	1	1	[ "A000012", "A001113", "A007953", "A021085", "A021733", "A392064" ]	null	Bruce Nye, Dec 29 2025	2026-01-04T12:13:55	oeisdata/seq/A392/A392064.seq	f3471ae514abaef1dd4d861d0ea1d2f5
A392065	a(n) = number of primes with unit exponent in the prime factorization of n, or 1 if no such primes exist.	[ "1", "1", "1", "1", "1", "2", "1", "1", "1", "2", "1", "1", "1", "2", "2", "1", "1", "1", "1", "1", "2", "2", "1", "1", "1", "2", "1", "1", "1", "3", "1", "1", "2", "2", "2", "1", "1", "2", "2", "1", "1", "3", "1", "1", "1", "...	[ "nonn", "changed" ]	13	1	6	[ "A056169", "A392065" ]	null	Paolo Xausa, Dec 30 2025	2026-01-06T18:02:00	oeisdata/seq/A392/A392065.seq	38626dce1ff7335f442dff076f6446e3
A392066	Irregular triangle read by rows: row n lists the primes with exponent > 1 in the prime factorization of n. If no such primes exist, row n = {0}.	[ "0", "0", "0", "2", "0", "0", "0", "2", "3", "0", "0", "2", "0", "0", "0", "2", "0", "3", "0", "2", "0", "0", "0", "2", "5", "0", "3", "2", "0", "0", "0", "2", "0", "0", "0", "2", "3", "0", "0", "0", "2", "0", "0", "0", "2", "...	[ "nonn", "tabf", "changed" ]	15	1	4	[ "A035306", "A056170", "A063958", "A071773", "A392066", "A392108", "A392109" ]	null	Paolo Xausa, Dec 30 2025	2026-01-07T03:09:42	oeisdata/seq/A392/A392066.seq	3daf35fea08d21a8993a215bd49fce08
A392068	a(n) = ((np)^p - 1)/(np-1) where p = prime(n).	[ "3", "43", "54241", "499738093", "257985813269495081", "51373494704737674143611", "1630858706046866494553223650456641", "1888214044824212617202713986685438034137", "898360659184642106918714737719916646001431214295057", "888472717356820461786099728280628417468328373702422145328719723183391" ]	[ "nonn", "easy", "new" ]	11	1	1	[ "A033286", "A392068" ]	null	Michel Lagneau, Dec 29 2025	2026-01-12T15:27:06	oeisdata/seq/A392/A392068.seq	87e11a0153d6e3d0e2e8f11003b90dc4
A392069	Powers k^m divisible by only 1 prime cube, with m > 1 and nonsquarefree weak k (in A332785).	[ "144", "324", "400", "576", "784", "1600", "1936", "2025", "2304", "2500", "2704", "2916", "3136", "3600", "3969", "4624", "5625", "5776", "6400", "7056", "7744", "8100", "8464", "9216", "9604", "9801", "10816", "12544", "13456", "13689", "14400", "15376...	[ "nonn", "easy", "new" ]	7	1	1	[ "A001597", "A001694", "A013929", "A024619", "A062503", "A126706", "A131605", "A286708", "A332785", "A359280", "A386762", "A389864", "A389947", "A391968", "A392069" ]	null	Michael De Vlieger, Jan 07 2026	2026-01-15T16:04:49	oeisdata/seq/A392/A392069.seq	8b7aae33537d58d13b9e07305b90fafa
A392070	Exponents k such that 2^k - 1 has exactly one representation of the form (2^k-1)/d + d with omega((2^k - 1)/d + d) = omega(2^k - 1) where omega = A001221.	[ "2", "3", "5", "6", "7", "9", "11", "13", "17", "18", "19", "25", "27", "29", "31", "32", "34", "43", "53", "55", "57", "61", "62", "65", "89", "93", "105", "107", "111", "115", "116", "117", "121", "127", "133", "138", "157", "158", "159",...	[ "nonn", "more", "new" ]	25	1	1	[ "A000043", "A000225", "A000668", "A001221", "A391451", "A392070" ]	null	Juri-Stepan Gerasimov, Dec 29 2025	2026-01-11T13:57:18	oeisdata/seq/A392/A392070.seq	ff4c743e83f0eeb0fc382db5af6d0917
A392074	a(n) = Sum_{k=0..floor(n/3)} 2^k * 3^(n-3k) binomial(k,4(n-3k)).	[ "1", "0", "0", "2", "0", "0", "4", "0", "0", "8", "0", "0", "16", "48", "0", "32", "480", "0", "64", "2880", "0", "128", "13440", "0", "256", "53760", "2304", "512", "193536", "41472", "1024", "645120", "414720", "2048", "2027520", "3041280", "...	[ "nonn" ]	17	0	4	[ "A178619", "A392042", "A392074" ]	null	Seiichi Manyama, Dec 30 2025	2025-12-31T08:49:18	oeisdata/seq/A392/A392074.seq	70ef367fd8aa296c4de574786ade0ca8
A392075	a(n) = Sum_{k=0..floor(n/3)} (k+1) * 2^k * 3^(n-3k) binomial(k,4(n-3k)).	[ "1", "0", "0", "4", "0", "0", "12", "0", "0", "32", "0", "0", "80", "240", "0", "192", "2880", "0", "448", "20160", "0", "1024", "107520", "0", "2304", "483840", "20736", "5120", "1935360", "414720", "11264", "7096320", "4561920", "24576", "2433024...	[ "nonn" ]	21	0	4	[ "A178619", "A392043", "A392075" ]	null	Seiichi Manyama, Dec 30 2025	2025-12-31T15:04:44	oeisdata/seq/A392/A392075.seq	1f45cb765362b0d72d946f1d9ad9fc54
A392076	a(n) = Sum_{k=0..n} (k+1) * binomial(k,4*(n-k)).	[ "1", "2", "3", "4", "5", "11", "37", "113", "289", "640", "1280", "2412", "4468", "8429", "16460", "33062", "67084", "135283", "268929", "526680", "1020344", "1966429", "3787730", "7309799", "14135460", "27353769", "52882135", "102017609", "196309933", "3768...	[ "nonn" ]	19	0	2	[ "A001629", "A005676", "A178619", "A391962", "A392044", "A392076" ]	null	Seiichi Manyama, Dec 30 2025	2025-12-31T04:14:52	oeisdata/seq/A392/A392076.seq	5d50cea6d8fcb3d107d2dde0728cb916
A392077	Numbers k such that sigma(k) = psi(k) + phi(k) + omega(k)^8.	[ "1275924", "2521260", "3035892", "4282824", "5956860", "9998004", "10790010", "11598930", "13432404", "15846500", "17399892", "19087284", "21014076", "21882324", "22994292", "25736620", "27230420", "28162134", "29593524", "30632980", "30768220", "32089540", "32176170", ...	[ "nonn" ]	8	1	1	[ "A000010", "A000203", "A001221", "A001615", "A389972", "A390049", "A391312", "A391749", "A391763", "A391908", "A391949", "A392077", "A392078" ]	null	S. I. Dimitrov, Dec 30 2025	2026-01-04T20:57:41	oeisdata/seq/A392/A392077.seq	b7f2fd136c878aad97aeac92f25aa4f8
A392078	Numbers k such that sigma(k) = psi(k) + phi(k) + omega(k)^9.	[ "786120", "7466772", "13995500", "14675968", "16055760", "18637080", "23941524", "41035260", "54282690", "55856680", "60484116", "61439124", "62574204", "68571090", "69111630", "70778964", "77774772", "96903876", "98065688", "103401100", "105680244", "108000084", "1116293...	[ "nonn" ]	13	1	1	[ "A000010", "A000203", "A001221", "A001615", "A389972", "A390049", "A391312", "A391749", "A391763", "A391908", "A391949", "A392077", "A392078" ]	null	S. I. Dimitrov, Dec 30 2025	2026-01-04T21:37:41	oeisdata/seq/A392/A392078.seq	9a97fca19ad3416b9ebc77e48fd1fe8e
A392079	a(n) is the maximum count of equal differences among consecutive divisors of A259366(n).	[ "2", "3", "2", "2", "2", "3", "3", "3", "2", "3", "2", "3", "2", "2", "5", "2", "2", "2", "4", "2", "2", "2", "4", "4", "2", "3", "2", "2", "2", "3", "3", "2", "2", "2", "5", "3", "2", "4", "2", "2", "2", "4", "3", "4", "2", "...	[ "nonn", "easy", "new" ]	14	1	1	[ "A000005", "A027750", "A060682", "A193829", "A259366", "A392079", "A392080" ]	null	Felix Huber, Jan 08 2026	2026-01-16T10:01:29	oeisdata/seq/A392/A392079.seq	ee920d3f963588f09b9d574df2f5722f
A392080	a(n) is the number of repeated difference values among the consecutive divisors of A259366(n).	[ "1", "1", "1", "2", "1", "2", "1", "2", "2", "2", "1", "3", "3", "1", "1", "1", "2", "1", "3", "2", "2", "2", "3", "2", "1", "4", "1", "2", "2", "1", "3", "1", "2", "2", "2", "3", "1", "2", "1", "2", "2", "3", "2", "2", "2", "...	[ "nonn", "new" ]	9	1	4	[ "A000005", "A027750", "A060682", "A193829", "A259366", "A392079", "A392080" ]	null	Felix Huber, Jan 09 2026	2026-01-16T10:01:24	oeisdata/seq/A392/A392080.seq	6c91e0d2751b2c6b482d739d9fee4443
A392082	Number of graceful Prüfer codes on n vertices with n-1 in the first position of the code.	[ "1", "2", "6", "18", "67", "301", "1550", "8850", "56682", "393858" ]	[ "nonn", "hard", "more", "new" ]	35	3	2	[ "A033472", "A392082" ]	null	Igor Blokhin, Dec 30 2025	2026-01-13T06:28:44	oeisdata/seq/A392/A392082.seq	a29da1863bd6c3582ab0c1e6c102494b
A392083	a(n) = 2*a(n-1) + 2^(n+1) - 1, a(0) = 0.	[ "0", "3", "13", "41", "113", "289", "705", "1665", "3841", "8705", "19457", "43009", "94209", "204801", "442369", "950273", "2031617", "4325377", "9175041", "19398657", "40894465", "85983233", "180355073", "377487361", "788529153", "1644167169", "3422552065", "7...	[ "nonn", "easy" ]	17	0	2	[ "A391958", "A392083" ]	null	A.H.M. Smeets, Dec 30 2025	2026-01-04T22:13:46	oeisdata/seq/A392/A392083.seq	8e689ed6b360b66c22d0b8f2c0e6bd23
A392084	a(n) = 3a(n-1) + 23^n - 1, a(0) = 0.	[ "0", "5", "32", "149", "608", "2309", "8384", "29525", "101696", "344453", "1151456", "3808661", "12488864", "40655237", "131531648", "423292757", "1355971712", "4326195461", "13753427360", "43584805013", "137727983840", "434104657925", "1365076092992", "4283514636629",...	[ "nonn", "easy", "new" ]	15	0	2	[ "A391959", "A392084" ]	null	A.H.M. Smeets, Dec 30 2025	2026-01-05T15:45:38	oeisdata/seq/A392/A392084.seq	ecc0980e2891c27da480cd84e7711aef
A392085	The unitary totient of the smallest number whose square is divisible by n.	[ "1", "1", "2", "1", "4", "2", "6", "3", "2", "4", "10", "2", "12", "6", "8", "3", "16", "2", "18", "4", "12", "10", "22", "6", "4", "12", "8", "6", "28", "8", "30", "7", "20", "16", "24", "2", "36", "18", "24", "12", "40", "12", "42...	[ "nonn", "mult", "easy" ]	8	1	3	[ "A002117", "A013661", "A019554", "A047994", "A183699", "A390752", "A392085" ]	null	Amiram Eldar, Dec 30 2025	2025-12-30T11:41:22	oeisdata/seq/A392/A392085.seq	3b12c2e5afd85d81b7f93c6ef8db1793
A392086	The unitary totient of the smallest number whose cube is divisible by n.	[ "1", "1", "2", "1", "4", "2", "6", "1", "2", "4", "10", "2", "12", "6", "8", "3", "16", "2", "18", "4", "12", "10", "22", "2", "4", "12", "2", "6", "28", "8", "30", "3", "20", "16", "24", "2", "36", "18", "24", "4", "40", "12", "42"...	[ "nonn", "mult", "easy" ]	9	1	3	[ "A013661", "A013663", "A019555", "A047994", "A390753", "A392086" ]	null	Amiram Eldar, Dec 30 2025	2025-12-30T11:41:18	oeisdata/seq/A392/A392086.seq	c386610dfda874c857333d7bd1ad572b
A392087	The unitary totient of the smallest square divisible by n.	[ "1", "3", "8", "3", "24", "24", "48", "15", "8", "72", "120", "24", "168", "144", "192", "15", "288", "24", "360", "72", "384", "360", "528", "120", "24", "504", "80", "144", "840", "576", "960", "63", "960", "864", "1152", "24", "1368", "108...	[ "nonn", "mult", "easy" ]	7	1	2	[ "A002117", "A013662", "A047994", "A053143", "A183700", "A327171", "A392087" ]	null	Amiram Eldar, Dec 30 2025	2025-12-30T11:41:25	oeisdata/seq/A392/A392087.seq	091719321de37a5e42eb15677496c740
A392088	The unitary totient of the smallest cube divisible by n.	[ "1", "7", "26", "7", "124", "182", "342", "7", "26", "868", "1330", "182", "2196", "2394", "3224", "63", "4912", "182", "6858", "868", "8892", "9310", "12166", "182", "124", "15372", "26", "2394", "24388", "22568", "29790", "63", "34580", "34384", ...	[ "nonn", "mult", "easy" ]	7	1	2	[ "A013662", "A013667", "A047994", "A053149", "A390754", "A392088" ]	null	Amiram Eldar, Dec 30 2025	2025-12-30T11:41:31	oeisdata/seq/A392/A392088.seq	c611e181c799c5797e89b383a92856a3
A392089	The unitary totient of the smallest powerful number that is divisible by n.	[ "1", "3", "8", "3", "24", "24", "48", "7", "8", "72", "120", "24", "168", "144", "192", "15", "288", "24", "360", "72", "384", "360", "528", "56", "24", "504", "26", "144", "840", "576", "960", "31", "960", "864", "1152", "24", "1368", "1080"...	[ "nonn", "mult", "easy" ]	8	1	2	[ "A001694", "A002117", "A013661", "A047994", "A183699", "A197863", "A330523", "A390755", "A392089" ]	null	Amiram Eldar, Dec 30 2025	2025-12-30T11:41:12	oeisdata/seq/A392/A392089.seq	ab1198e298f1c95267216b2b44e75f4d
A392090	The unitary totient of the smallest cubefull number that is divisible by n.	[ "1", "7", "26", "7", "124", "182", "342", "7", "26", "868", "1330", "182", "2196", "2394", "3224", "15", "4912", "182", "6858", "868", "8892", "9310", "12166", "182", "124", "15372", "26", "2394", "24388", "22568", "29790", "31", "34580", "34384", ...	[ "nonn", "mult", "easy" ]	8	1	2	[ "A002117", "A013662", "A036966", "A047994", "A063453", "A356193", "A390756", "A392090" ]	null	Amiram Eldar, Dec 30 2025	2025-12-30T11:41:08	oeisdata/seq/A392/A392090.seq	d3182e8fc42b5c1c63b807a1198fde4c
A392095	Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + Sum_{j=0..k} A(n-1,j)*A(k-j,0) with A(0,k) = 1.	[ "1", "1", "2", "1", "4", "6", "1", "10", "18", "24", "1", "34", "64", "94", "118", "1", "152", "278", "414", "556", "674", "1", "826", "1438", "2096", "2842", "3634", "4308", "1", "5134", "8528", "12046", "16110", "20772", "25754", "30062", "1"...	[ "nonn", "tabl" ]	4	0	3	[ "A088713", "A392095" ]	null	Mikhail Kurkov, Dec 30 2025	2026-01-04T21:07:20	oeisdata/seq/A392/A392095.seq	faff64b9df96ec7f40761b6f40a13e1d
A392096	Numbers k for which the width pattern of the symmetric representation of sigma for k, SRS(k), has the form 1 2 1 0 1 2 1 0 . . . 0 1 2 1 0 1 2 1 containing at least one 0.	[ "78", "102", "114", "138", "174", "186", "222", "246", "258", "282", "318", "348", "354", "366", "372", "402", "426", "438", "444", "474", "492", "498", "516", "534", "564", "582", "606", "618", "636", "642", "654", "678", "708", "732", "762", "7...	[ "nonn", "new" ]	12	1	1	[ "A003056", "A237270", "A237271", "A237593", "A249223", "A249351", "A367377", "A370205", "A377654", "A392096" ]	null	Hartmut F. W. Hoft, Dec 30 2025	2026-01-06T15:29:10	oeisdata/seq/A392/A392096.seq	15e0c76ecb29e3e8ead3fb6aa9aa8ef3
A392097	a(n) is the smallest number k for which the symmetric representation of sigma for k, SRS(k), has n parts and each i-th part up to the diagonal is unimodal of width i.	[ "1", "3", "15", "75", "1225", "1275", "2145", "43125" ]	[ "nonn", "more", "new" ]	9	1	2	[ "A003056", "A237270", "A237271", "A237591", "A237593", "A249223", "A249351", "A377654", "A392097" ]	null	Hartmut F. W. Hoft, Dec 30 2025	2026-01-11T19:21:29	oeisdata/seq/A392/A392097.seq	2c6d82f1547f046adcc982c9e773ddc9
A392098	Powers k^m divisible by at least 2 cubes greater than 1, with m > 1 and nonsquarefree weak k (in A332785).	[ "1728", "5832", "8000", "13824", "20736", "21952", "32400", "63504", "64000", "85184", "90000", "91125", "104976", "110592", "125000", "129600", "140608", "156816", "157464", "160000", "175616", "202500", "216000", "219024", "248832", "250047", "254016", "291600...	[ "nonn", "easy", "new" ]	8	1	1	[ "A001597", "A001694", "A013929", "A024619", "A062503", "A126706", "A131605", "A286708", "A332785", "A359280", "A386762", "A389864", "A391968", "A392069", "A392098" ]	null	Michael De Vlieger, Jan 08 2026	2026-01-14T17:48:11	oeisdata/seq/A392/A392098.seq	a3f32d0b6a3d9127be016ebb7b40f1dc
A392099	Smallest prime p such that p + 3^n - 1 is also a prime.	[ "2", "3", "3", "3", "3", "29", "5", "17", "3", "5", "3", "71", "17", "17", "3", "3", "29", "89", "11", "137", "47", "167", "5", "107", "3", "29", "3", "17", "71", "131", "5", "17", "71", "71", "59", "23", "3", "197", "179", "83", "233",...	[ "nonn", "new" ]	21	0	1	[ "A000040", "A001359", "A051783", "A056206", "A342502", "A392099" ]	null	James S. DeArmon, Dec 30 2025	2026-01-08T22:20:21	oeisdata/seq/A392/A392099.seq	04a2ee731a3ff13627a8b8320e882dbd
A392100	Array listed by rows: Pythagorean triples a < b < c such that a + b and a + b + 2*c are prime, sorted first by increasing perimeter and then by the even term.	[ "3", "4", "5", "5", "12", "13", "28", "45", "53", "11", "60", "61", "36", "77", "85", "65", "72", "97", "15", "112", "113", "24", "143", "145", "88", "105", "137", "119", "120", "169", "28", "195", "197", "21", "220", "221", "60", "221", "2...	[ "nonn", "tabf", "new" ]	8	1	1	[ "A103605", "A103606", "A392100" ]	null	Will Gosnell and Robert Israel, Dec 30 2025	2026-01-05T15:16:56	oeisdata/seq/A392/A392100.seq	eefece70248efa3ee267e31813f4ade1
A392101	Numbers whose square is the concatenation of two or more powers of 2.	[ "9", "11", "12", "18", "21", "22", "29", "38", "41", "46", "91", "92", "108", "109", "111", "119", "121", "122", "129", "168", "204", "208", "221", "338", "341", "358", "359", "378", "404", "408", "471", "478", "482", "642", "696", "804", "908"...	[ "nonn", "base", "new" ]	59	1	1	[ "A272884", "A392101", "A392102" ]	null	Rhys Feltman, Dec 30 2025	2026-01-12T13:59:40	oeisdata/seq/A392/A392101.seq	4ef89783fff664213a24b722f7e26b6e
A392102	Squares which are the concatenation of two or more powers of 2.	[ "81", "121", "144", "324", "441", "484", "841", "1444", "1681", "2116", "8281", "8464", "11664", "11881", "12321", "14161", "14641", "14884", "16641", "28224", "41616", "43264", "48841", "114244", "116281", "128164", "128881", "142884", "163216", "166464", ...	[ "nonn", "base", "new" ]	68	1	1	[ "A000290", "A272884", "A381259", "A392101", "A392102" ]	null	Rhys Feltman, Dec 30 2025	2026-01-12T16:06:40	oeisdata/seq/A392/A392102.seq	13dfd47c37b33d22f054bd2e33c97d39
A392104	a(n) = prime(A391796(n)).	[ "5", "23", "47", "251", "1889", "7793", "43451", "243161", "726893", "759821", "1820123", "1820111", "10141499", "19725479", "19725473", "136209239", "400414127", "400414121", "489144659", "489144599", "766319201", "766319189", "21549657581", "21549657551", "215496575...	[ "nonn", "new" ]	16	1	1	[ "A391796", "A391807", "A392104" ]	null	Clark Kimberling, Jan 08 2026	2026-01-17T12:56:33	oeisdata/seq/A392/A392104.seq	c2509ad83b84a5bef72472d1bffdfd17
A392106	Concatenation of binary words using letters 1 and 2, in numerical order.	[ "1", "2", "1", "1", "1", "2", "2", "1", "2", "2", "1", "1", "1", "1", "1", "2", "1", "2", "1", "1", "2", "2", "2", "1", "1", "2", "1", "2", "2", "2", "1", "2", "2", "2", "1", "1", "1", "1", "1", "1", "1", "2", "1", "1", "2", "...	[ "nonn", "easy", "new" ]	11	0	2	[ "A007931", "A076478", "A392106", "A392107" ]	null	Clark Kimberling, Jan 09 2026	2026-01-14T19:23:54	oeisdata/seq/A392/A392106.seq	6ae25c7e1e584b9d157d3e1f0f7ecc88
A392108	a(n) = smallest prime with exponent > 1 in the prime factorization of n, or 0 if no such prime exists.	[ "0", "0", "0", "2", "0", "0", "0", "2", "3", "0", "0", "2", "0", "0", "0", "2", "0", "3", "0", "2", "0", "0", "0", "2", "5", "0", "3", "2", "0", "0", "0", "2", "0", "0", "0", "2", "0", "0", "0", "2", "0", "0", "0", "2", "3", "...	[ "nonn", "changed" ]	19	1	4	[ "A056170", "A063958", "A080368", "A392066", "A392108", "A392109" ]	null	Paolo Xausa, Dec 31 2025	2026-01-07T03:09:20	oeisdata/seq/A392/A392108.seq	319d0dcc2f8b79e79d96ea02c6320075
A392109	a(n) = largest prime with exponent > 1 in the prime factorization of n, or 0 if no such prime exists.	[ "0", "0", "0", "2", "0", "0", "0", "2", "3", "0", "0", "2", "0", "0", "0", "2", "0", "3", "0", "2", "0", "0", "0", "2", "5", "0", "3", "2", "0", "0", "0", "2", "0", "0", "0", "3", "0", "0", "0", "2", "0", "0", "0", "2", "3", "...	[ "nonn", "changed" ]	23	1	4	[ "A027748", "A056170", "A063958", "A080367", "A392066", "A392108", "A392109" ]	null	Paolo Xausa, Dec 31 2025	2026-01-17T23:10:46	oeisdata/seq/A392/A392109.seq	f2de5133abb0b232fe537cffe8f2c239
A392110	Numbers k such that 5k - (greatest prime < 5n) < (least prime > 5n) - 5k.	[ "4", "5", "11", "15", "16", "17", "18", "22", "23", "28", "35", "37", "40", "43", "46", "47", "49", "53", "57", "59", "64", "68", "70", "71", "72", "75", "76", "77", "78", "82", "85", "87", "88", "89", "90", "94", "96", "100", "101", "102...	[ "nonn", "new" ]	15	1	1	[ "A000027", "A000040", "A390788", "A392110", "A392111", "A392112" ]	null	Clark Kimberling, Jan 11 2026	2026-01-15T13:08:17	oeisdata/seq/A392/A392110.seq	c2331c40b36c5fbb8b97c73b7e851772
A392111	Numbers k such that 5k - (greatest prime < 5n) = (least prime > 5n) - 5k.	[ "1", "3", "6", "9", "10", "12", "21", "24", "30", "32", "33", "34", "36", "39", "41", "45", "48", "52", "54", "60", "63", "74", "81", "84", "93", "99", "103", "112", "114", "118", "120", "122", "123", "125", "129", "130", "132", "135", "136...	[ "nonn", "new" ]	8	1	2	[ "A000027", "A000040", "A390788", "A392110", "A392111", "A392112" ]	null	Clark Kimberling, Jan 15 2026	2026-01-18T23:43:47	oeisdata/seq/A392/A392111.seq	87b6466d51065f2862ad06102c55d9f7
A392112	Numbers k such that 5k - (greatest prime < 5n) > (least prime > 5n) - 5k.	[ "2", "7", "8", "13", "14", "19", "20", "25", "26", "27", "29", "31", "38", "42", "44", "50", "51", "55", "56", "58", "61", "62", "65", "66", "67", "69", "73", "79", "80", "83", "86", "91", "92", "95", "97", "98", "104", "107", "108", "109...	[ "nonn", "new" ]	8	1	1	[ "A000027", "A000040", "A390788", "A392110", "A392111", "A392112" ]	null	Clark Kimberling, Jan 15 2026	2026-01-18T23:43:23	oeisdata/seq/A392/A392112.seq	f1cae2134574a1cac780239967c70917
A392113	Numbers k such that 6k - (greatest prime < 6n) < (least prime > 6n) - 6k.	[ "4", "8", "9", "14", "15", "19", "22", "28", "29", "34", "36", "39", "42", "43", "44", "49", "53", "59", "60", "64", "65", "67", "74", "75", "78", "80", "82", "84", "85", "88", "93", "94", "98", "99", "104", "108", "109", "111", "113", "1...	[ "nonn", "new" ]	10	1	1	[ "A000027", "A000040", "A390788", "A392113", "A392114", "A392115" ]	null	Clark Kimberling, Jan 15 2026	2026-01-18T14:28:54	oeisdata/seq/A392/A392113.seq	48e57bc8cc77931094bd8899654069ae
A392114	Numbers k such that 6k - (greatest prime < 6n) = (least prime > 6n) - 6k.	[ "1", "2", "3", "5", "7", "10", "12", "17", "18", "20", "23", "24", "25", "30", "31", "32", "33", "38", "40", "41", "45", "47", "48", "50", "52", "54", "57", "58", "69", "70", "71", "72", "77", "87", "92", "95", "97", "100", "103", "106", ...	[ "nonn", "new" ]	6	1	2	[ "A000027", "A000040", "A390788", "A392113", "A392114", "A392115" ]	null	Clark Kimberling, Jan 16 2026	2026-01-18T14:29:18	oeisdata/seq/A392/A392114.seq	3f37103fb6c96ad31ce7f26cbb0d9ad2
A392115	Numbers k such that 6k - (greatest prime < 6n) > (least prime > 6n) - 6k.	[ "6", "11", "13", "16", "21", "26", "27", "35", "37", "46", "51", "55", "56", "61", "62", "63", "66", "68", "73", "76", "79", "81", "83", "86", "89", "90", "91", "96", "101", "102", "105", "112", "115", "118", "121", "122", "123", "125", "12...	[ "nonn", "new" ]	6	1	1	[ "A000027", "A000040", "A390788", "A392113", "A392114", "A392115" ]	null	Clark Kimberling, Jan 16 2026	2026-01-18T14:29:43	oeisdata/seq/A392/A392115.seq	50e5077b42ea688f80b5b0293fb68f70
A392126	The smallest k >= 0 that can be represented as a linear combination of 1^3, 2^3, ..., n^3 with coefficients +-1 and that cannot be represented using 1^3, 2^3, ..., m^3 with 1<=m<n.	[ "1", "7", "18", "28", "25", "45", "26", "4", "5", "15", "12", "0", "3", "13", "8", "24", "83", "4463", "6932", "10710", "9269", "27845", "30410", "50660", "47853", "80109", "64610", "117368", "104237", "163709", "139880", "220500", "190709", "289205"...	[ "nonn", "new" ]	19	1	2	[ "A000537", "A000578", "A019568", "A392025", "A392126", "A392127", "A392346" ]	null	Alois P. Heinz, Dec 30 2025	2026-01-08T18:53:12	oeisdata/seq/A392/A392126.seq	355fcb66462060000d57892ec570c65c
A392127	The smallest k >= 0 that can be represented as a linear combination of 1^2, 2^2, ..., n^2 with coefficients +-1 and that cannot be represented using 1^2, 2^2, ..., m^2 with 1<=m<n.	[ "1", "3", "4", "2", "13", "7", "0", "44", "25", "61", "52", "250", "241", "563", "458", "984", "823", "1529", "1272", "2214", "1853", "3055", "2614", "4068", "3539", "5269", "4644", "6674", "5945", "8299", "7458", "10160", "9199", "12273", "11184",...	[ "nonn", "new" ]	21	1	2	[ "A000290", "A000330", "A019568", "A231015", "A392126", "A392127", "A392346" ]	null	Alois P. Heinz, Dec 30 2025	2026-01-08T18:53:06	oeisdata/seq/A392/A392127.seq	56e45a03a105b476421cbea13d97d2ac
A392129	Integers k such that there are i groups of order k+i up to isomorphism, for 1 <= i <= 6.	[ "29436120", "117873720", "222954120", "324510120", "689250312", "876688920", "894332520", "946770120", "1417704120", "1761885720", "1834771320", "2199390120", "2240912520", "2509602120", "2779308072", "3091545720", "3143256120", "3262502520", "3436558920", "3585632472", "3652...	[ "nonn", "new" ]	22	1	1	[ "A000001", "A003277", "A054395", "A054396", "A054397", "A055561", "A135850", "A249550", "A373648", "A373649", "A373650", "A381335", "A392129", "A392130" ]	null	Gabriel Eiseman, Dec 31 2025	2026-01-06T22:02:43	oeisdata/seq/A392/A392129.seq	a22021b4d11f45d958bbf27312e64268
A392130	Integers k such that there are i groups of order k+1 up to isomorphism, for i=1,2,3,4,5,6,7.	[ "5973822114120", "6305771634120", "28058687347320", "48414128744520", "74556478687320", "84300170172120", "142366076070120", "153432090884520", "207916418382120", "339858901929960", "375278016786120", "415728077892120", "426239950426920" ]	[ "nonn", "more", "new" ]	12	1	1	[ "A000001", "A003277", "A054395", "A054396", "A054397", "A055561", "A135850", "A249550", "A373648", "A373649", "A373650", "A381335", "A392129", "A392130" ]	null	Gabriel Eiseman, Dec 31 2025	2026-01-12T16:37:11	oeisdata/seq/A392/A392130.seq	a330c792b7a238cbf32a203861d61d46
A392131	a(n) is the determinant of the n X n circulant matrix whose rows are formed by successively rotating the vector (A000129(0), A000129(1), ..., A000129(n-1)) right.	[ "1", "0", "1", "9", "640", "252100", "597791376", "8252767236301", "665993168101048320", "313398861092499186897912", "859858277306561684298748471552", "13751722491910335588624048111158497953", "1281922118247965546737056182821273600000000000", "6965087592028412199949848226918542065689144193...	[ "nonn" ]	10	0	4	[ "A000129", "A086459", "A118713", "A318173", "A392131", "A392132" ]	null	Stefano Spezia, Dec 31 2025	2026-01-02T09:31:36	oeisdata/seq/A392/A392131.seq	6846cafb2b0608c70f3f5e8405a1612f
A392132	a(n) is the permanent of the n X n circulant matrix whose rows are formed by successively rotating the vector (A000129(0), A000129(1), ..., A000129(n-1)) right.	[ "1", "0", "1", "9", "772", "317660", "755353604", "10245351511241", "806176992721352256", "369503021786486764387032", "989096947940706813698038153600", "15475597969523884739005106748390084993", "1415492646595459083986339197781974314319509760", "756726561532204931161254789744755112027328520...	[ "nonn" ]	8	0	4	[ "A000129", "A306457", "A392131", "A392132" ]	null	Stefano Spezia, Dec 31 2025	2026-01-02T09:31:45	oeisdata/seq/A392/A392132.seq	32a26ee5dc176bcba30b155410e72ec1
A392135	Third column of A391957.	[ "0", "0", "0", "0", "0", "9", "7", "5", "3", "1", "18", "14", "10", "6", "2", "27", "21", "15", "9", "3", "36", "28", "20", "12", "4", "94", "82", "70", "58", "46", "93", "79", "65", "51", "37", "92", "76", "60", "44", "28", "91", "73...	[ "nonn", "new" ]	37	0	6	[ "A055457", "A391956", "A391957", "A392135" ]	null	A.H.M. Smeets, Jan 01 2026	2026-01-13T13:40:20	oeisdata/seq/A392/A392135.seq	9758a6c0df3f5cd9bbec89da84f3ecd5
A392136	Fourth column of A391957.	[ "0", "0", "0", "0", "0", "0", "0", "13", "11", "9", "7", "5", "3", "1", "26", "22", "18", "14", "10", "6", "2", "39", "33", "27", "21", "15", "9", "3", "52", "44", "36", "28", "20", "12", "4", "65", "55", "45", "35", "25", "15", "5", ...	[ "nonn", "new" ]	21	0	8	[ "A373217", "A391956", "A391957", "A392136" ]	null	A.H.M. Smeets, Jan 05 2026	2026-01-15T17:53:24	oeisdata/seq/A392/A392136.seq	dda68005ba65f536e8fb0aa6f56b7791
A392137	a(n) = 5a(n-1) + 25^n - 1, a(0) = 0.	[ "0", "9", "94", "719", "4844", "30469", "183594", "1074219", "6152344", "34667969", "192871094", "1062011719", "5798339844", "31433105469", "169372558594", "907897949219", "4844665527344", "25749206542969", "136375427246094", "720024108886719", "3790855407714844", "19907951...	[ "nonn", "easy", "new" ]	21	0	2	[ "A392135", "A392137" ]	null	A.H.M. Smeets, Jan 05 2026	2026-01-15T09:39:05	oeisdata/seq/A392/A392137.seq	4c5e6793c85203a733c081f0fe437f9b
A392138	a(n) = 7a(n-1) + 27^n - 1, a(0) = 0.	[ "0", "13", "188", "2001", "18808", "165269", "1392180", "11392345", "91276016", "719639325", "5602425772", "43171633889", "329884011624", "2502966102181", "18877208860964", "141635585046633", "1057914956465632", "7870665723233837", "58351487258457756", "431258201179950577", "...	[ "nonn", "easy", "new" ]	14	0	2	[ "A392136", "A392138" ]	null	A.H.M. Smeets, Jan 06 2026	2026-01-13T07:58:53	oeisdata/seq/A392/A392138.seq	b7212f73526252213e9aea684a8671a7

A392000

Array read by antidiagonals: A(k, n) is the number of undirected closed Hamiltonian paths on the k X n knight graph.

[ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "...

[ "nonn", "tabl" ]

15

1

50

[ "A000004", "A001230", "A169764", "A175855", "A175881", "A193054", "A193055", "A390833", "A391999", "A392000" ]

null

Peter Luschny, Dec 26 2025

2025-12-27T13:09:56

oeisdata/seq/A392/A392000.seq

6e9b9973be577742bc33e83c3672430d

A392001

Numbers k such that k + 1 is not a power of an odd prime and 2*k + 1 is not a prime power.

[ "7", "17", "19", "25", "27", "31", "32", "34", "37", "38", "43", "45", "47", "49", "55", "57", "59", "61", "64", "67", "71", "73", "76", "77", "79", "85", "87", "91", "92", "93", "94", "97", "101", "103", "104", "107", "109", "110", "115", ...

[ "nonn", "changed" ]

25

1

[ "A003418", "A045979", "A053176", "A067656", "A093880", "A166602", "A246655", "A392001" ]

null

Peter Luschny, Dec 31 2025

2026-01-17T20:15:06

oeisdata/seq/A392/A392001.seq

6b6e535814cd96b3c06595971d76a4b7

A392002

Number of partitions of n with parts colored by {0, 1} such that the sum of colors is congruent to 1 (mod 2).

[ "0", "1", "2", "5", "9", "18", "31", "55", "90", "150", "237", "376", "577", "885", "1325", "1978", "2900", "4235", "6100", "8745", "12400", "17501", "24477", "34075", "47079", "64756", "88493", "120420", "162940", "219595", "294476", "393407", "523237...

[ "nonn" ]

11

0

3

[ "A000041", "A000712", "A387957", "A390208", "A392002" ]

null

Thomas Hutton, Dec 26 2025

2026-01-01T17:33:09

oeisdata/seq/A392/A392002.seq

db28f4c59031632aa50add244f6f47e6

A392003

a(n) = (7*2^n - 9 + 5*(-1)^n) / 6.

[ "0", "4", "7", "18", "35", "74", "147", "298", "595", "1194", "2387", "4778", "9555", "19114", "38227", "76458", "152915", "305834", "611667", "1223338", "2446675", "4893354", "9786707", "19573418", "39146835", "78293674", "156587347", "313174698", "626349395"...

[ "nonn", "easy" ]

30

1

2

[ "A266753", "A390768", "A392003" ]

null

Karl-Heinz Hofmann, Dec 26 2025

2026-01-02T08:29:06

oeisdata/seq/A392/A392003.seq

b6f8d82f63e5c44fca18d9f2ba373369

A392004

Numbers k such that the base-k digits of 2^k include at least one 0.

[ "2", "4", "8", "10", "16", "19", "31", "32", "35", "39", "46", "48", "61", "64", "71", "93", "102", "103", "104", "108", "109", "117", "122", "128", "139", "154", "158", "166", "171", "181", "184", "185", "192", "193", "199", "218", "226", "2...

[ "nonn", "base" ]

18

1

[ "A068535", "A392004" ]

null

Robert Israel, Dec 26 2025

2025-12-30T11:41:50

oeisdata/seq/A392/A392004.seq

10e136873c24ae7b8acdb8f6a078bb02

A392007

Smallest number k such that log_b(k) - floor(log_b(k)) >= 1/2 for b = 2..n, and not for b = n+1.

[ "3", "6", "47", "56", "920483", "46341", "888667668", "13207922282067" ]

[ "nonn", "more", "new" ]

50

2

1

[ "A002163", "A002193", "A002194", "A010464", "A020857", "A153459", "A392007" ]

null

Jakub Buczak, Dec 26 2025

2026-01-05T14:05:15

oeisdata/seq/A392/A392007.seq

fe9019a5879694cd0616a242824c2083

A392008

Odd squarefree integers with at least 3 prime factors.

[ "105", "165", "195", "231", "255", "273", "285", "345", "357", "385", "399", "429", "435", "455", "465", "483", "555", "561", "595", "609", "615", "627", "645", "651", "663", "665", "705", "715", "741", "759", "777", "795", "805", "861", "885", "...

[ "nonn", "changed" ]

45

1

[ "A001221", "A001222", "A046389", "A350352", "A392008", "A392009" ]

null

Charles Kusniec, Dec 26 2025

2026-01-08T10:10:41

oeisdata/seq/A392/A392008.seq

bd1123b73f1431610754232c118d53ea

A392009

Even squarefree integers with at least 3 prime factors.

[ "30", "42", "66", "70", "78", "102", "110", "114", "130", "138", "154", "170", "174", "182", "186", "190", "210", "222", "230", "238", "246", "258", "266", "282", "286", "290", "310", "318", "322", "330", "354", "366", "370", "374", "390", "402",...

[ "nonn", "changed" ]

36

1

[ "A001221", "A001222", "A053858", "A075819", "A350352", "A392008", "A392009" ]

null

Charles Kusniec, Dec 26 2025

2026-01-08T09:40:57

oeisdata/seq/A392/A392009.seq

134d74c1ae92b07ff47cfd3a101f9963

A392012

The sum of the exponential divisors of the cubefree numbers.

[ "1", "2", "3", "6", "5", "6", "7", "12", "10", "11", "18", "13", "14", "15", "17", "24", "19", "30", "21", "22", "23", "30", "26", "42", "29", "30", "31", "33", "34", "35", "72", "37", "38", "39", "41", "42", "43", "66", "60", "46", "47...

[ "nonn", "easy" ]

11

1

2

[ "A002117", "A004709", "A013662", "A051377", "A057723", "A322791", "A322857", "A382419", "A392012" ]

null

Amiram Eldar, Dec 27 2025

2025-12-28T09:57:15

oeisdata/seq/A392/A392012.seq

15b5c7eabc32b6e02f29eb138023a83e

A392013

The number of exponential divisors of the exponentially-2^n numbers.

[ "1", "1", "1", "2", "1", "1", "1", "2", "1", "1", "2", "1", "1", "1", "3", "1", "2", "1", "2", "1", "1", "1", "2", "1", "2", "1", "1", "1", "1", "1", "1", "4", "1", "1", "1", "1", "1", "1", "2", "2", "1", "1", "3", "2", "2", "...

[ "nonn", "easy" ]

8

1

4

[ "A049419", "A138302", "A271727", "A322791", "A392013", "A392014", "A392015" ]

null

Amiram Eldar, Dec 27 2025

2025-12-28T09:57:23

oeisdata/seq/A392/A392013.seq

ce4813de9e394438741ec9178f03b6d9

A392014

The number of exponential unitary divisors of the exponentially-2^n numbers.

[ "1", "1", "1", "2", "1", "1", "1", "2", "1", "1", "2", "1", "1", "1", "2", "1", "2", "1", "2", "1", "1", "1", "2", "1", "2", "1", "1", "1", "1", "1", "1", "4", "1", "1", "1", "1", "1", "1", "2", "2", "1", "1", "2", "2", "2", "...

[ "nonn", "easy" ]

8

1

4

[ "A138302", "A278908", "A361255", "A366538", "A392013", "A392014", "A392016" ]

null

Amiram Eldar, Dec 27 2025

2025-12-28T09:57:18

oeisdata/seq/A392/A392014.seq

3000ca0ddfc68ba54c3e279cbcd45f9c

A392015

The sum of the exponential divisors of the exponentially-2^n numbers.

[ "1", "1", "1", "3", "1", "1", "1", "4", "1", "1", "3", "1", "1", "1", "7", "1", "4", "1", "3", "1", "1", "1", "6", "1", "3", "1", "1", "1", "1", "1", "1", "12", "1", "1", "1", "1", "1", "1", "3", "4", "1", "1", "7", "8", "6", ...

[ "nonn", "easy" ]

8

1

4

[ "A051377", "A138302", "A322791", "A392013", "A392015", "A392016" ]

null

Amiram Eldar, Dec 27 2025

2025-12-28T09:57:27

oeisdata/seq/A392/A392015.seq

dd315e2e827ab5bfe18ad24120d01fdd

A392016

The sum of the exponential unitary divisors of the exponentially-2^n numbers.

[ "1", "2", "3", "6", "5", "6", "7", "12", "10", "11", "18", "13", "14", "15", "18", "17", "24", "19", "30", "21", "22", "23", "30", "26", "42", "29", "30", "31", "33", "34", "35", "72", "37", "38", "39", "41", "42", "43", "66", "60", "46...

[ "nonn", "easy" ]

8

1

2

[ "A138302", "A271727", "A322857", "A361255", "A392014", "A392015", "A392016" ]

null

Amiram Eldar, Dec 27 2025

2025-12-28T09:57:31

oeisdata/seq/A392/A392016.seq

139c433d8c7361fcd7732f53a2237110

A392017

Decimal expansion of sqrt(3)*log(2+sqrt(3)).

[ "2", "2", "8", "1", "0", "3", "7", "9", "8", "8", "9", "0", "2", "8", "3", "9", "0", "4", "2", "5", "9", "3", "2", "8", "2", "7", "6", "4", "6", "4", "1", "2", "1", "7", "4", "7", "2", "6", "0", "4", "8", "8", "3", "1", "9", "...

[ "nonn", "cons", "easy" ]

24

1

[ "A065918", "A196530", "A257436", "A259830", "A392017" ]

null

Artur Jasinski, Dec 27 2025

2026-01-01T17:57:37

oeisdata/seq/A392/A392017.seq

70eb036ddf9d1020fb77dcbae2b22aeb

A392019

If binomial(n,k) = u*v where the only primes dividing u are <= k and the only primes dividing v are > k, then a(n) is the least k such that u > n^2, or 0 if such a k doesn't exist.

[ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "8", "0", "0", "0", "12", "13", "13", "5", "0", "7", "0", "9", "12", "11", "11", "7", "11", "17", "15", "14", "17", "17", "19", "1...

[ "nonn" ]

12

1

21

[ "A007318", "A392019" ]

null

Elijah Beregovsky, Dec 27 2025

2026-01-01T22:16:27

oeisdata/seq/A392/A392019.seq

50439b8533cc1193d3e8768986cad290

A392020

Expansion of the series e(-q), where the series e(q) is Morier-Genoud and Ovsienko's q-analog of Euler's number e.

[ "1", "-1", "0", "-1", "0", "1", "2", "3", "3", "1", "-3", "-9", "-17", "-25", "-29", "-23", "2", "54", "134", "232", "320", "347", "243", "-71", "-660", "-1531", "-2575", "-3504", "-3804", "-2747", "488", "6537", "15395", "25819", "34716", "36780...

[ "sign", "easy", "new" ]

28

0

7

[ "A001113", "A003417", "A004148", "A337589", "A392020", "A392021", "A392022", "A392023" ]

null

Peter Bala, Dec 27 2025

2026-01-11T09:20:57

oeisdata/seq/A392/A392020.seq

9c7f73d23b203212143951a52f5d3385

A392021

Expansion of the series 1/e(-q), where the series e(q) is Morier-Genoud and Ovsienko's q-analog of the real number e = exp(1).

[ "1", "1", "1", "2", "3", "3", "2", "-1", "-8", "-20", "-37", "-57", "-71", "-59", "12", "190", "531", "1073", "1785", "2476", "2657", "1381", "-2877", "-12163", "-28550", "-52828", "-81924", "-104671", "-96015", "-10924", "218528", "675902", "1431270",...

[ "sign", "easy", "new" ]

18

0

4

[ "A001113", "A003417", "A337589", "A392020", "A392021", "A392022", "A392023" ]

null

Peter Bala, Jan 04 2026

2026-01-11T09:21:01

oeisdata/seq/A392/A392021.seq

c45d191c4145e4f7b65a60f983a03b99

A392022

Expansion of the series e(q)*e(-q) (even coefficients only), where the series e(q) is Morier-Genoud and Ovsienko's q-analog of the real number e = exp(1).

[ "1", "-1", "-2", "5", "14", "1", "-52", "-125", "-83", "347", "1217", "1579", "-949", "-8529", "-17399", "-10159", "40670", "134560", "167114", "-104686", "-858254", "-1591711", "-453531", "4997043", "13433932", "12368077", "-22722489", "-102188424", "-1567129...

[ "sign", "easy", "new" ]

17

0

3

[ "A001113", "A003417", "A337589", "A392020", "A392021", "A392022", "A392023" ]

null

Peter Bala, Jan 05 2026

2026-01-11T09:21:06

oeisdata/seq/A392/A392022.seq

6ecb30bb0cab3799ae9f1d10009aa54e

A392023

Expansion of the series e(q)/e(-q), where the series e(q) is Morier-Genoud and Ovsienko's q-analog of the real number e = exp(1).

[ "1", "2", "2", "4", "6", "6", "8", "2", "-6", "-26", "-62", "-96", "-156", "-150", "-106", "150", "698", "1590", "3142", "4630", "6234", "5436", "528", "-12876", "-40696", "-83918", "-148010", "-210998", "-248122", "-175238", "126894", "823188", "21380...

[ "sign", "easy", "new" ]

13

0

2

[ "A001113", "A003417", "A337589", "A392020", "A392021", "A392022", "A392023" ]

null

Peter Bala, Jan 06 2026

2026-01-11T09:21:10

oeisdata/seq/A392/A392023.seq

dc71d90f5d45f5123dac00eb0b64a89b

A392025

Least m such that n can be represented as a linear combination of 1^3, 2^3, ..., m^3 with coefficients +-1 only.

[ "12", "1", "12", "13", "8", "9", "8", "2", "15", "2", "15", "9", "11", "14", "11", "10", "11", "10", "3", "10", "3", "13", "8", "13", "16", "5", "7", "5", "4", "14", "4", "13", "7", "10", "3", "10", "3", "13", "11", "10", "11", "5", ...

[ "nonn", "look" ]

23

0

1

[ "A231015", "A392025" ]

null

Jeffrey Shallit, Dec 27 2025

2025-12-31T22:20:54

oeisdata/seq/A392/A392025.seq

bcc95d620ef0cec77d12f923e21dc7dc

A392026

Decimal expansion of 113*Pi + 355.

[ "7", "0", "9", "9", "9", "9", "9", "6", "9", "8", "5", "5", "6", "4", "6", "6", "3", "5", "9", "4", "6", "2", "7", "8", "7", "0", "2", "3", "1", "0", "5", "8", "3", "8", "2", "5", "9", "1", "4", "2", "8", "0", "1", "4", "2", "...

[ "nonn", "cons" ]

26

3

1

[ "A014307", "A068079", "A180875", "A226043", "A392026" ]

null

Artur Jasinski, Dec 27 2025

2026-01-02T13:52:24

oeisdata/seq/A392/A392026.seq

976ece7e1c2a789f2eeaefeff38cc0b5

A392027

Partial sums of A255933.

[ "1", "3", "5", "8", "10", "13", "15", "18", "20", "23", "27", "29", "31", "33", "35", "38", "40", "43", "45", "48", "50", "53", "55", "58", "60", "63", "65", "68", "70", "73", "75", "78", "80", "83", "85", "88", "90", "93", "95", "98", ...

[ "nonn" ]

11

1

2

[ "A255933", "A392027", "A392028" ]

null

Paolo Xausa, Dec 27 2025

2025-12-28T09:54:48

oeisdata/seq/A392/A392027.seq

d888708f6a40512cdbedff7d5086865c

A392028

Indices of records in A255933.

[ "1", "2", "4", "11", "49", "286", "1997", "31925", "287310", "1436559", "31604222", "189625346", "4930259198", "34511814535" ]

[ "nonn", "hard", "more" ]

14

1

2

[ "A255933", "A392027", "A392028" ]

null

Paolo Xausa, Dec 27 2025

2025-12-29T12:25:58

oeisdata/seq/A392/A392028.seq

108ee240f8e4509bec605fccc43505fa

A392030

Numbers k with squarefree kernel 210 such that both k-1 and k+1 are prime.

[ "420", "1050", "3360", "5880", "6300", "7350", "7560", "8820", "10500", "26250", "26880", "28350", "29400", "30870", "33600", "35280", "100800", "102060", "131250", "134400", "158760", "170100", "241920", "264600", "282240", "294000", "370440", "420000", "4536...

[ "nonn", "easy", "new" ]

6

1

[ "A002110", "A007947", "A014574", "A027856", "A147571", "A386252", "A392030" ]

null

Michael De Vlieger, Jan 06 2026

2026-01-08T09:28:01

oeisdata/seq/A392/A392030.seq

0457871d6219aa5537c5b66f382bed2d

A392031

a(n) is the smallest k which has exactly n divisors d such that sigma(d) = sigma(d + k/d) where sigma = A000203.

[ "1", "14", "120", "4080", "132132", "7361280" ]

[ "nonn", "more" ]

12

0

2

[ "A000005", "A000203", "A392031" ]

null

Juri-Stepan Gerasimov, Dec 27 2025

2026-01-04T12:14:13

oeisdata/seq/A392/A392031.seq

8007a1e3e024f952ee5164e7cd696f05

A392032

a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(n,k) * binomial(n-k,k).

[ "1", "1", "7", "19", "73", "241", "831", "2787", "9339", "31003", "102373", "336073", "1097911", "3570607", "11565529", "37324437", "120052107", "384962859", "1230978093", "3926097681", "12492150819", "39660238779", "125656218933", "397360559841", "1254333690477", "...

[ "nonn" ]

14

0

3

[ "A002426", "A098473", "A391990", "A392032", "A392033" ]

null

Seiichi Manyama, Dec 27 2025

2025-12-31T14:26:13

oeisdata/seq/A392/A392032.seq

d51229e9b48b86f850024859b68f09af

A392033

a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(n,k) * binomial(n-k,k).

[ "1", "1", "9", "25", "109", "381", "1421", "5069", "18075", "63499", "221323", "764523", "2621587", "8928115", "30220347", "101721707", "340659227", "1135543323", "3769023795", "12460643043", "41045661279", "134748546639", "440974812831", "1438893866655", "46822262130...

[ "nonn" ]

15

0

3

[ "A002426", "A098473", "A391990", "A392032", "A392033" ]

null

Seiichi Manyama, Dec 27 2025

2025-12-31T12:34:33

oeisdata/seq/A392/A392033.seq

c622caefa74a66a3436986aada26f278

A392034

a(n) = prime(n)*(prime(n+1) - prime(n))/2.

[ "1", "3", "5", "14", "11", "26", "17", "38", "69", "29", "93", "74", "41", "86", "141", "159", "59", "183", "134", "71", "219", "158", "249", "356", "194", "101", "206", "107", "218", "791", "254", "393", "137", "695", "149", "453", "471", "3...

[ "nonn", "look", "new" ]

26

1

2

[ "A000040", "A001223", "A291463", "A392034" ]

null

Anthony S. Wright, Dec 27 2025

2026-01-10T20:26:38

oeisdata/seq/A392/A392034.seq

f90f1ca0700f401a176e73e2dc3736b0

A392035

Irregular table read by rows: the n-th row contains, in weakly decreasing order, the positive integers whose squares sum to n as obtained by the greedy algorithm.

[ "1", "1", "1", "1", "1", "1", "2", "2", "1", "2", "1", "1", "2", "1", "1", "1", "2", "2", "3", "3", "1", "3", "1", "1", "3", "1", "1", "1", "3", "2", "3", "2", "1", "3", "2", "1", "1", "4", "4", "1", "4", "1", "1", "4", "1", "...

[ "nonn", "tabf" ]

10

1

7

[ "A000196", "A053610", "A350674", "A392035" ]

null

Stefano Spezia, Dec 28 2025

2025-12-29T07:33:59

oeisdata/seq/A392/A392035.seq

d37e20c69416e913cdbcbce9c83272d1

A392036

Irregular table read by rows: the n-th row contains, in weakly decreasing order, the powers of 2 summing to n as obtained by the greedy algorithm.

[ "1", "2", "2", "1", "4", "4", "1", "4", "2", "4", "2", "1", "8", "8", "1", "8", "2", "8", "2", "1", "8", "4", "8", "4", "1", "8", "4", "2", "8", "4", "2", "1", "16", "16", "1", "16", "2", "16", "2", "1", "16", "4", "16", "4", "1...

[ "nonn", "tabf" ]

12

1

2

[ "A000079", "A000120", "A006519", "A053644", "A272011", "A392036" ]

null

Stefano Spezia, Dec 28 2025

2026-01-02T09:31:16

oeisdata/seq/A392/A392036.seq

db152c379c30cbe616b8c21bb77a2dfe

A392037

Square table read by downward antidiagonals: n-th row has e.g.f. (1-9*x)^(-n/9).

[ "1", "0", "1", "0", "1", "1", "0", "10", "2", "1", "0", "190", "22", "3", "1", "0", "5320", "440", "36", "4", "1", "0", "196840", "12760", "756", "52", "5", "1", "0", "9054640", "484880", "22680", "1144", "70", "6", "1", "0", "498005200", "...

[ "nonn", "easy", "tabl" ]

9

0

8

[ "A035012", "A035013", "A035017", "A035018", "A035020", "A035021", "A035022", "A035023", "A045756", "A049211", "A051232", "A053116", "A084949", "A144758", "A144829", "A147629", "A147630", "A147631", "A392037" ]

null

Jianing Song, Dec 29 2025

2025-12-29T07:31:21

oeisdata/seq/A392/A392037.seq

614f6b0cc0553c469aec24c970803712

A392038

Least number k such that n can be represented as a linear combination of 1^n, 2^n, ..., k^n with coefficients +-1 only.

[ "2", "1", "4", "13", "15", "25", "31", "25", "47", "57", "60" ]

[ "nonn", "more", "new" ]

59

0

1

[ "A019568", "A126908", "A231015", "A392025", "A392038" ]

null

Jean-Marc Rebert, Dec 29 2025

2026-01-17T17:23:15

oeisdata/seq/A392/A392038.seq

a30159461b9e51b5ab77021df019f227

A392041

Smallest prime number whose decimal digits include the consecutive pattern 1, 2, ..., n.

[ "11", "127", "1123", "12343", "123457", "12345623", "71234567", "1234567801", "1234567891", "1234567891003", "12345678910111", "12345678910111207", "912345678910111213", "123456789101112131449", "12345678910111213141523", "123456789101112131415161", "41234567891011121314151617", "3...

[ "nonn", "base" ]

27

1

[ "A007908", "A053546", "A058183", "A176942", "A392041" ]

null

Jean-Marc Rebert, Dec 29 2025

2026-01-04T13:32:47

oeisdata/seq/A392/A392041.seq

ab5361b59d64affdae698b4eb12b4e37

A392042

a(n) = Sum_{k=0..floor(n/3)} 2^k * 3^(n-3*k) * binomial(k,3*(n-3*k)).

[ "1", "0", "0", "2", "0", "0", "4", "0", "0", "8", "24", "0", "16", "192", "0", "32", "960", "0", "64", "3840", "576", "128", "13440", "8064", "256", "43008", "64512", "512", "129024", "387072", "14848", "368640", "1935360", "278528", "1013760", "...

[ "nonn" ]

18

0

4

[ "A178618", "A392042", "A392043" ]

null

Seiichi Manyama, Dec 29 2025

2025-12-31T08:49:15

oeisdata/seq/A392/A392042.seq

fb45a442cfe2483df0d3d40f7a6583e7

A392043

a(n) = Sum_{k=0..floor(n/3)} (k+1) * 2^k * 3^(n-3*k) * binomial(k,3*(n-3*k)).

[ "1", "0", "0", "4", "0", "0", "12", "0", "0", "32", "96", "0", "80", "960", "0", "192", "5760", "0", "448", "26880", "4032", "1024", "107520", "64512", "2304", "387072", "580608", "5120", "1290240", "3870720", "149504", "4055040", "21288960", "306585...

[ "nonn" ]

18

0

4

[ "A178618", "A392042", "A392043" ]

null

Seiichi Manyama, Dec 29 2025

2025-12-31T08:49:10

oeisdata/seq/A392/A392043.seq

9cedcf924a08bc8f5b2dc16b9e84521b

A392044

a(n) = Sum_{k=0..n} (k+1) * binomial(k,3*(n-k)).

[ "1", "2", "3", "4", "9", "26", "67", "148", "296", "570", "1103", "2172", "4313", "8528", "16691", "32360", "62365", "119830", "229840", "440052", "840653", "1602038", "3045967", "5779676", "10947997", "20706438", "39107943", "73764072", "138955192", "261449...

[ "nonn" ]

19

0

2

[ "A001629", "A003522", "A178618", "A391962", "A392044" ]

null

Seiichi Manyama, Dec 29 2025

2025-12-31T12:34:42

oeisdata/seq/A392/A392044.seq

da9cdce0caf9d12e8a5409e65a2d9106

A392045

Odd positive integers whose canonical prime factorization p_1^e_1*...*p_r^e_r satisfies that {p_1, ..., p_r, e_1, ..., e_r} is a set of 2*r consecutive integers.

[ "9", "81", "625", "2025", "5625", "15625", "117649", "455625", "1265625", "5764801", "22325625", "37515625", "43758225", "62015625", "73530625", "238239225", "337640625", "661775625", "25937424601", "45956640625", "90075015625", "246140015625", "482434430625", "68372226...

[ "nonn", "easy", "new" ]

8

1

[ "A097320", "A111774", "A382330", "A383397", "A387172", "A387586", "A389339", "A389340", "A389795", "A390784", "A392045", "A392046", "A392047" ]

null

Felix Huber, Jan 05 2026

2026-01-08T23:14:53

oeisdata/seq/A392/A392045.seq

f04bf9b0f8db9dc0e5824e83f4f484e6

A392046

Even positive integers whose canonical prime factorization p_1^e_1*...*p_r^e_r satisfies that {p_1, ..., p_r, e_1, ..., e_r} is a set of 2*r consecutive integers.

[ "2", "8", "48", "162", "2000", "2592", "3888", "5000", "25920", "58320", "120000", "750000", "911250", "2531250", "13720000", "19208000", "37340352", "56010528", "58320000", "85750000", "87127488", "87480000", "162000000", "196036848", "235298000", "300125000", "3...

[ "nonn", "easy", "fini", "full", "new" ]

8

1

[ "A097320", "A111774", "A382330", "A383397", "A387172", "A387586", "A389339", "A389340", "A389795", "A390784", "A392045", "A392046", "A392047" ]

null

Felix Huber, Jan 05 2026

2026-01-09T16:08:13

oeisdata/seq/A392/A392046.seq

1d3ba6c371482a1cf937e0ded265e87c

A392047

Positive integers whose canonical prime factorization p_1^e_1*...*p_r^e_r satisfies that {p_1, ..., p_r, e_1, ..., e_r} is a set of 2*r consecutive integers.

[ "2", "8", "9", "48", "81", "162", "625", "2000", "2025", "2592", "3888", "5000", "5625", "15625", "25920", "58320", "117649", "120000", "455625", "750000", "911250", "1265625", "2531250", "5764801", "13720000", "19208000", "22325625", "37340352", "37515625", ...

[ "nonn", "easy", "new" ]

12

1

[ "A097320", "A111774", "A382330", "A383397", "A387172", "A387586", "A389339", "A389340", "A389795", "A390784", "A392045", "A392046", "A392047" ]

null

Felix Huber, Jan 05 2026

2026-01-09T16:13:09

oeisdata/seq/A392/A392047.seq

e1330bf8981a16a047798fbafd61ed81

A392048

a(n) is the number of distinct integers that can be created by summing combinations of the n+1 first rationals {b(0)/b(1), ..., b(n)/b(n+1)}, where b(k)=A002487(k) is Stern's diatomic series.

[ "1", "2", "2", "4", "4", "6", "7", "10", "10", "11", "11", "13", "14", "16", "17", "21", "21", "22", "22", "24", "24", "25", "25", "29", "30", "32", "33", "36", "37", "39", "40", "45", "45", "46", "46", "48", "48", "49", "49", "53", "53...

[ "nonn", "new" ]

11

0

2

[ "A002487", "A392048" ]

null

Frederik G. Faye, Dec 29 2025

2026-01-04T22:32:51

oeisdata/seq/A392/A392048.seq

7c79a22ef47c8805d8b2c0d9c7b91e1a

A392049

a(n) is the numerator of the harmonic mean of the digits of n.

[ "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "0", "1", "4", "3", "8", "5", "12", "7", "16", "9", "0", "4", "2", "12", "8", "20", "3", "28", "16", "36", "0", "3", "12", "3", "24", "15", "4", "21", "48", "9", "0", "8", "8", "24...

[ "nonn", "base", "easy", "frac" ]

11

0

3

[ "A011540", "A371383", "A392049", "A392050", "A392051" ]

null

Stefano Spezia, Dec 29 2025

2026-01-02T09:28:07

oeisdata/seq/A392/A392049.seq

fb37ae93deefd359387f173f620db166

A392050

a(n) is the denominator of the harmonic mean of the digits of n.

[ "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "3", "2", "5", "3", "7", "4", "9", "5", "1", "3", "1", "5", "3", "7", "1", "9", "5", "11", "1", "2", "5", "1", "7", "4", "1", "5", "11", "2", "1", "5", "3", "7", "1", ...

[ "nonn", "base", "easy", "frac" ]

8

0

13

[ "A011540", "A371384", "A392049", "A392050", "A392051" ]

null

Stefano Spezia, Dec 29 2025

2026-01-02T09:30:43

oeisdata/seq/A392/A392050.seq

0df9db7e1af64d222d4eb808bb562878

A392051

Nonnegative integers such that the harmonic mean of the digits is an integer.

[ "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "20", "22", "26", "30", "33", "36", "40", "44", "50", "55", "60", "62", "63", "66", "70", "77", "80", "88", "90", "99", "100", "101", "102", "103", "104", "105", "106", "107", ...

[ "nonn", "base", "easy" ]

8

1

3

[ "A011540", "A061383", "A061430", "A147591", "A392049", "A392050", "A392051" ]

null

Stefano Spezia, Dec 29 2025

2026-01-02T09:29:36

oeisdata/seq/A392/A392051.seq

e1e0ccb342569bbbe41ff69c1b3ec505

A392055

a(n) = p if n = p^k is a perfect power of either a ramified or inert prime in the Gaussian integers and 1 otherwise.

[ "1", "2", "3", "2", "1", "1", "7", "2", "3", "1", "11", "1", "1", "1", "1", "2", "1", "1", "19", "1", "1", "1", "23", "1", "1", "1", "3", "1", "1", "1", "31", "2", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "43", "1", "1"...

[ "nonn", "easy", "new" ]

28

1

2

[ "A002145", "A014963", "A045326", "A392055", "A392059" ]

null

Rick Zhou, Dec 29 2025

2026-01-05T16:21:34

oeisdata/seq/A392/A392055.seq

e76cae950dfebfa533c3840626cf4649

A392056

Numbers k such that (47^k - 4^k)/43 is prime.

[ "7", "12703", "15679", "49711", "90749" ]

[ "nonn", "hard", "more" ]

5

1

[ "A062587", "A062589", "A127996", "A127997", "A128344", "A204940", "A217320", "A225807", "A229542", "A375161", "A375236", "A377031", "A392056" ]

null

Robert Price, Dec 29 2025

2025-12-30T13:14:14

oeisdata/seq/A392/A392056.seq

4133511f4c6160678a3727ccd5d72b03

A392057

a(0)=0, a(n) = wt(a(n-1)) + wt(n), where wt(x) is the binary weight of x, A000120.

[ "0", "1", "2", "3", "3", "4", "3", "5", "3", "4", "3", "5", "4", "4", "4", "5", "3", "4", "3", "5", "4", "4", "4", "5", "4", "4", "4", "5", "5", "6", "6", "7", "4", "3", "4", "4", "3", "5", "5", "6", "4", "4", "4", "5", "5", "...

[ "nonn", "base", "easy", "new" ]

17

0

3

[ "A000120", "A000788", "A392057" ]

null

Alex Ratushnyak, Dec 29 2025

2026-01-10T20:30:05

oeisdata/seq/A392/A392057.seq

f32e1f38a8bbeb76cf2db3a867c60ebb

A392058

Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + Sum_{j=0..k} binomial(k+1,j)*A(n-1,j)*A(k-j,0) with A(0,k) = 1.

[ "1", "1", "2", "1", "5", "7", "1", "17", "31", "38", "1", "83", "178", "254", "292", "1", "578", "1330", "2099", "2683", "2975", "1", "5474", "12748", "21338", "29425", "35375", "38350", "1", "66932", "153136", "263870", "382966", "489383", "566083...

[ "nonn", "tabl" ]

6

0

3

[ "A233335", "A392058" ]

null

Mikhail Kurkov, Dec 29 2025

2026-01-04T20:28:11

oeisdata/seq/A392/A392058.seq

50abcfbcb5ec38ac470e802fc4d753d0

A392059

a(n) is the product of all p^k such that p is a ramified or inert prime in the Gaussian integers and k is the largest such k satisfying p^k <= n.

[ "1", "2", "6", "12", "12", "12", "84", "168", "504", "504", "5544", "5544", "5544", "5544", "5544", "11088", "11088", "11088", "210672", "210672", "210672", "210672", "4845456", "4845456", "4845456", "4845456", "14536368", "14536368", "14536368", "14536368",...

[ "nonn", "look", "easy", "new" ]

17

1

2

[ "A387569", "A392055", "A392059" ]

null

Rick Zhou, Dec 29 2025

2026-01-10T23:01:37

oeisdata/seq/A392/A392059.seq

a857d99b3e377a04e6b5c2f973766a5f

A392061

Irregular triangular array read by rows: T(n,k) is the number of periodic functions on [n] that have order k, n>=0, 1<=k<=A000793(n).

[ "1", "1", "3", "1", "10", "9", "2", "41", "69", "32", "6", "196", "555", "360", "150", "24", "20", "1057", "4845", "3680", "2520", "864", "840", "6322", "45843", "37800", "37590", "19152", "22050", "720", "0", "0", "504", "0", "420", "41393", "...

[ "nonn", "tabf", "look", "changed" ]

30

0

3

[ "A000248", "A000793", "A006153", "A057731", "A162682", "A199673", "A222029", "A392061" ]

null

Geoffrey Critzer, Dec 29 2025

2026-01-05T15:10:42

oeisdata/seq/A392/A392061.seq

7334f3ddc714a7d6e56eeab265c5e7d1

A392062

Numbers k such that every prime factor of the sum of decimal digits of k is greater than every prime factor of k.

[ "1", "10", "16", "25", "32", "49", "56", "98", "100", "128", "160", "175", "245", "250", "256", "289", "490", "539", "560", "595", "616", "625", "676", "728", "748", "784", "845", "847", "896", "968", "980", "1000", "1024", "1280", "1372", "1408"...

[ "nonn", "base" ]

10

1

2

[ "A006530", "A007953", "A011557", "A020639", "A392062" ]

null

Robert Israel, Dec 29 2025

2026-01-04T20:33:28

oeisdata/seq/A392/A392062.seq

5203faa7962e8427d9fef000db806016

A392064

Decimal expansion of the expected number of consecutive random decimal digits whose running sum first exceeds 9.

[ "2", "8", "6", "7", "9", "7", "1", "9", "9", "0", "7", "9", "2", "4", "4", "1", "3", "1", "3", "3", "2", "2", "2", "5", "7", "2", "3", "1", "2", "4", "0", "8", "3", "6", "9", "0", "6", "5", "6", "6", "1", "6", "5", "4", "0", "...

[ "nonn", "cons", "easy" ]

44

1

[ "A000012", "A001113", "A007953", "A021085", "A021733", "A392064" ]

null

Bruce Nye, Dec 29 2025

2026-01-04T12:13:55

oeisdata/seq/A392/A392064.seq

f3471ae514abaef1dd4d861d0ea1d2f5

A392065

a(n) = number of primes with unit exponent in the prime factorization of n, or 1 if no such primes exist.

[ "1", "1", "1", "1", "1", "2", "1", "1", "1", "2", "1", "1", "1", "2", "2", "1", "1", "1", "1", "1", "2", "2", "1", "1", "1", "2", "1", "1", "1", "3", "1", "1", "2", "2", "2", "1", "1", "2", "2", "1", "1", "3", "1", "1", "1", "...

[ "nonn", "changed" ]

13

1

6

[ "A056169", "A392065" ]

null

Paolo Xausa, Dec 30 2025

2026-01-06T18:02:00

oeisdata/seq/A392/A392065.seq

38626dce1ff7335f442dff076f6446e3

A392066

Irregular triangle read by rows: row n lists the primes with exponent > 1 in the prime factorization of n. If no such primes exist, row n = {0}.

[ "0", "0", "0", "2", "0", "0", "0", "2", "3", "0", "0", "2", "0", "0", "0", "2", "0", "3", "0", "2", "0", "0", "0", "2", "5", "0", "3", "2", "0", "0", "0", "2", "0", "0", "0", "2", "3", "0", "0", "0", "2", "0", "0", "0", "2", "...

[ "nonn", "tabf", "changed" ]

15

1

4

[ "A035306", "A056170", "A063958", "A071773", "A392066", "A392108", "A392109" ]

null

Paolo Xausa, Dec 30 2025

2026-01-07T03:09:42

oeisdata/seq/A392/A392066.seq

3daf35fea08d21a8993a215bd49fce08

A392068

a(n) = ((n*p)^p - 1)/(n*p-1) where p = prime(n).

[ "3", "43", "54241", "499738093", "257985813269495081", "51373494704737674143611", "1630858706046866494553223650456641", "1888214044824212617202713986685438034137", "898360659184642106918714737719916646001431214295057", "888472717356820461786099728280628417468328373702422145328719723183391" ]

[ "nonn", "easy", "new" ]

11

1

[ "A033286", "A392068" ]

null

Michel Lagneau, Dec 29 2025

2026-01-12T15:27:06

oeisdata/seq/A392/A392068.seq

87e11a0153d6e3d0e2e8f11003b90dc4

A392069

Powers k^m divisible by only 1 prime cube, with m > 1 and nonsquarefree weak k (in A332785).

[ "144", "324", "400", "576", "784", "1600", "1936", "2025", "2304", "2500", "2704", "2916", "3136", "3600", "3969", "4624", "5625", "5776", "6400", "7056", "7744", "8100", "8464", "9216", "9604", "9801", "10816", "12544", "13456", "13689", "14400", "15376...

[ "nonn", "easy", "new" ]

7

1

[ "A001597", "A001694", "A013929", "A024619", "A062503", "A126706", "A131605", "A286708", "A332785", "A359280", "A386762", "A389864", "A389947", "A391968", "A392069" ]

null

Michael De Vlieger, Jan 07 2026

2026-01-15T16:04:49

oeisdata/seq/A392/A392069.seq

8b7aae33537d58d13b9e07305b90fafa

A392070

Exponents k such that 2^k - 1 has exactly one representation of the form (2^k-1)/d + d with omega((2^k - 1)/d + d) = omega(2^k - 1) where omega = A001221.

[ "2", "3", "5", "6", "7", "9", "11", "13", "17", "18", "19", "25", "27", "29", "31", "32", "34", "43", "53", "55", "57", "61", "62", "65", "89", "93", "105", "107", "111", "115", "116", "117", "121", "127", "133", "138", "157", "158", "159",...

[ "nonn", "more", "new" ]

25

1

[ "A000043", "A000225", "A000668", "A001221", "A391451", "A392070" ]

null

Juri-Stepan Gerasimov, Dec 29 2025

2026-01-11T13:57:18

oeisdata/seq/A392/A392070.seq

ff4c743e83f0eeb0fc382db5af6d0917

A392074

a(n) = Sum_{k=0..floor(n/3)} 2^k * 3^(n-3*k) * binomial(k,4*(n-3*k)).

[ "1", "0", "0", "2", "0", "0", "4", "0", "0", "8", "0", "0", "16", "48", "0", "32", "480", "0", "64", "2880", "0", "128", "13440", "0", "256", "53760", "2304", "512", "193536", "41472", "1024", "645120", "414720", "2048", "2027520", "3041280", "...

[ "nonn" ]

17

0

4

[ "A178619", "A392042", "A392074" ]

null

Seiichi Manyama, Dec 30 2025

2025-12-31T08:49:18

oeisdata/seq/A392/A392074.seq

70ef367fd8aa296c4de574786ade0ca8

A392075

a(n) = Sum_{k=0..floor(n/3)} (k+1) * 2^k * 3^(n-3*k) * binomial(k,4*(n-3*k)).

[ "1", "0", "0", "4", "0", "0", "12", "0", "0", "32", "0", "0", "80", "240", "0", "192", "2880", "0", "448", "20160", "0", "1024", "107520", "0", "2304", "483840", "20736", "5120", "1935360", "414720", "11264", "7096320", "4561920", "24576", "2433024...

[ "nonn" ]

21

0

4

[ "A178619", "A392043", "A392075" ]

null

Seiichi Manyama, Dec 30 2025

2025-12-31T15:04:44

oeisdata/seq/A392/A392075.seq

1f45cb765362b0d72d946f1d9ad9fc54

A392076

a(n) = Sum_{k=0..n} (k+1) * binomial(k,4*(n-k)).

[ "1", "2", "3", "4", "5", "11", "37", "113", "289", "640", "1280", "2412", "4468", "8429", "16460", "33062", "67084", "135283", "268929", "526680", "1020344", "1966429", "3787730", "7309799", "14135460", "27353769", "52882135", "102017609", "196309933", "3768...

[ "nonn" ]

19

0

2

[ "A001629", "A005676", "A178619", "A391962", "A392044", "A392076" ]

null

Seiichi Manyama, Dec 30 2025

2025-12-31T04:14:52

oeisdata/seq/A392/A392076.seq

5d50cea6d8fcb3d107d2dde0728cb916

A392077

Numbers k such that sigma(k) = psi(k) + phi(k) + omega(k)^8.

[ "1275924", "2521260", "3035892", "4282824", "5956860", "9998004", "10790010", "11598930", "13432404", "15846500", "17399892", "19087284", "21014076", "21882324", "22994292", "25736620", "27230420", "28162134", "29593524", "30632980", "30768220", "32089540", "32176170", ...

[ "nonn" ]

8

1

[ "A000010", "A000203", "A001221", "A001615", "A389972", "A390049", "A391312", "A391749", "A391763", "A391908", "A391949", "A392077", "A392078" ]

null

S. I. Dimitrov, Dec 30 2025

2026-01-04T20:57:41

oeisdata/seq/A392/A392077.seq

b7f2fd136c878aad97aeac92f25aa4f8

A392078

Numbers k such that sigma(k) = psi(k) + phi(k) + omega(k)^9.

[ "786120", "7466772", "13995500", "14675968", "16055760", "18637080", "23941524", "41035260", "54282690", "55856680", "60484116", "61439124", "62574204", "68571090", "69111630", "70778964", "77774772", "96903876", "98065688", "103401100", "105680244", "108000084", "1116293...

[ "nonn" ]

13

1

[ "A000010", "A000203", "A001221", "A001615", "A389972", "A390049", "A391312", "A391749", "A391763", "A391908", "A391949", "A392077", "A392078" ]

null

S. I. Dimitrov, Dec 30 2025

2026-01-04T21:37:41

oeisdata/seq/A392/A392078.seq

9a97fca19ad3416b9ebc77e48fd1fe8e

A392079

a(n) is the maximum count of equal differences among consecutive divisors of A259366(n).

[ "2", "3", "2", "2", "2", "3", "3", "3", "2", "3", "2", "3", "2", "2", "5", "2", "2", "2", "4", "2", "2", "2", "4", "4", "2", "3", "2", "2", "2", "3", "3", "2", "2", "2", "5", "3", "2", "4", "2", "2", "2", "4", "3", "4", "2", "...

[ "nonn", "easy", "new" ]

14

1

[ "A000005", "A027750", "A060682", "A193829", "A259366", "A392079", "A392080" ]

null

Felix Huber, Jan 08 2026

2026-01-16T10:01:29

oeisdata/seq/A392/A392079.seq

ee920d3f963588f09b9d574df2f5722f

A392080

a(n) is the number of repeated difference values among the consecutive divisors of A259366(n).

[ "1", "1", "1", "2", "1", "2", "1", "2", "2", "2", "1", "3", "3", "1", "1", "1", "2", "1", "3", "2", "2", "2", "3", "2", "1", "4", "1", "2", "2", "1", "3", "1", "2", "2", "2", "3", "1", "2", "1", "2", "2", "3", "2", "2", "2", "...

[ "nonn", "new" ]

9

1

4

[ "A000005", "A027750", "A060682", "A193829", "A259366", "A392079", "A392080" ]

null

Felix Huber, Jan 09 2026

2026-01-16T10:01:24

oeisdata/seq/A392/A392080.seq

6c91e0d2751b2c6b482d739d9fee4443

A392082

Number of graceful Prüfer codes on n vertices with n-1 in the first position of the code.

[ "1", "2", "6", "18", "67", "301", "1550", "8850", "56682", "393858" ]

[ "nonn", "hard", "more", "new" ]

35

3

2

[ "A033472", "A392082" ]

null

Igor Blokhin, Dec 30 2025

2026-01-13T06:28:44

oeisdata/seq/A392/A392082.seq

a29da1863bd6c3582ab0c1e6c102494b

A392083

a(n) = 2*a(n-1) + 2^(n+1) - 1, a(0) = 0.

[ "0", "3", "13", "41", "113", "289", "705", "1665", "3841", "8705", "19457", "43009", "94209", "204801", "442369", "950273", "2031617", "4325377", "9175041", "19398657", "40894465", "85983233", "180355073", "377487361", "788529153", "1644167169", "3422552065", "7...

[ "nonn", "easy" ]

17

0

2

[ "A391958", "A392083" ]

null

A.H.M. Smeets, Dec 30 2025

2026-01-04T22:13:46

oeisdata/seq/A392/A392083.seq

8e689ed6b360b66c22d0b8f2c0e6bd23

A392084

a(n) = 3*a(n-1) + 2*3^n - 1, a(0) = 0.

[ "0", "5", "32", "149", "608", "2309", "8384", "29525", "101696", "344453", "1151456", "3808661", "12488864", "40655237", "131531648", "423292757", "1355971712", "4326195461", "13753427360", "43584805013", "137727983840", "434104657925", "1365076092992", "4283514636629",...

[ "nonn", "easy", "new" ]

15

0

2

[ "A391959", "A392084" ]

null

A.H.M. Smeets, Dec 30 2025

2026-01-05T15:45:38

oeisdata/seq/A392/A392084.seq

ecc0980e2891c27da480cd84e7711aef

A392085

The unitary totient of the smallest number whose square is divisible by n.

[ "1", "1", "2", "1", "4", "2", "6", "3", "2", "4", "10", "2", "12", "6", "8", "3", "16", "2", "18", "4", "12", "10", "22", "6", "4", "12", "8", "6", "28", "8", "30", "7", "20", "16", "24", "2", "36", "18", "24", "12", "40", "12", "42...

[ "nonn", "mult", "easy" ]

8

1

3

[ "A002117", "A013661", "A019554", "A047994", "A183699", "A390752", "A392085" ]

null

Amiram Eldar, Dec 30 2025

2025-12-30T11:41:22

oeisdata/seq/A392/A392085.seq

3b12c2e5afd85d81b7f93c6ef8db1793

A392086

The unitary totient of the smallest number whose cube is divisible by n.

[ "1", "1", "2", "1", "4", "2", "6", "1", "2", "4", "10", "2", "12", "6", "8", "3", "16", "2", "18", "4", "12", "10", "22", "2", "4", "12", "2", "6", "28", "8", "30", "3", "20", "16", "24", "2", "36", "18", "24", "4", "40", "12", "42"...

[ "nonn", "mult", "easy" ]

9

1

3

[ "A013661", "A013663", "A019555", "A047994", "A390753", "A392086" ]

null

Amiram Eldar, Dec 30 2025

2025-12-30T11:41:18

oeisdata/seq/A392/A392086.seq

c386610dfda874c857333d7bd1ad572b

A392087

The unitary totient of the smallest square divisible by n.

[ "1", "3", "8", "3", "24", "24", "48", "15", "8", "72", "120", "24", "168", "144", "192", "15", "288", "24", "360", "72", "384", "360", "528", "120", "24", "504", "80", "144", "840", "576", "960", "63", "960", "864", "1152", "24", "1368", "108...

[ "nonn", "mult", "easy" ]

7

1

2

[ "A002117", "A013662", "A047994", "A053143", "A183700", "A327171", "A392087" ]

null

Amiram Eldar, Dec 30 2025

2025-12-30T11:41:25

oeisdata/seq/A392/A392087.seq

091719321de37a5e42eb15677496c740

A392088

The unitary totient of the smallest cube divisible by n.

[ "1", "7", "26", "7", "124", "182", "342", "7", "26", "868", "1330", "182", "2196", "2394", "3224", "63", "4912", "182", "6858", "868", "8892", "9310", "12166", "182", "124", "15372", "26", "2394", "24388", "22568", "29790", "63", "34580", "34384", ...

[ "nonn", "mult", "easy" ]

7

1

2

[ "A013662", "A013667", "A047994", "A053149", "A390754", "A392088" ]

null

Amiram Eldar, Dec 30 2025

2025-12-30T11:41:31

oeisdata/seq/A392/A392088.seq

c611e181c799c5797e89b383a92856a3

A392089

The unitary totient of the smallest powerful number that is divisible by n.

[ "1", "3", "8", "3", "24", "24", "48", "7", "8", "72", "120", "24", "168", "144", "192", "15", "288", "24", "360", "72", "384", "360", "528", "56", "24", "504", "26", "144", "840", "576", "960", "31", "960", "864", "1152", "24", "1368", "1080"...

[ "nonn", "mult", "easy" ]

8

1

2

[ "A001694", "A002117", "A013661", "A047994", "A183699", "A197863", "A330523", "A390755", "A392089" ]

null

Amiram Eldar, Dec 30 2025

2025-12-30T11:41:12

oeisdata/seq/A392/A392089.seq

ab1198e298f1c95267216b2b44e75f4d

A392090

The unitary totient of the smallest cubefull number that is divisible by n.

[ "1", "7", "26", "7", "124", "182", "342", "7", "26", "868", "1330", "182", "2196", "2394", "3224", "15", "4912", "182", "6858", "868", "8892", "9310", "12166", "182", "124", "15372", "26", "2394", "24388", "22568", "29790", "31", "34580", "34384", ...

[ "nonn", "mult", "easy" ]

8

1

2

[ "A002117", "A013662", "A036966", "A047994", "A063453", "A356193", "A390756", "A392090" ]

null

Amiram Eldar, Dec 30 2025

2025-12-30T11:41:08

oeisdata/seq/A392/A392090.seq

d3182e8fc42b5c1c63b807a1198fde4c

A392095

Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + Sum_{j=0..k} A(n-1,j)*A(k-j,0) with A(0,k) = 1.

[ "1", "1", "2", "1", "4", "6", "1", "10", "18", "24", "1", "34", "64", "94", "118", "1", "152", "278", "414", "556", "674", "1", "826", "1438", "2096", "2842", "3634", "4308", "1", "5134", "8528", "12046", "16110", "20772", "25754", "30062", "1"...

[ "nonn", "tabl" ]

4

0

3

[ "A088713", "A392095" ]

null

Mikhail Kurkov, Dec 30 2025

2026-01-04T21:07:20

oeisdata/seq/A392/A392095.seq

faff64b9df96ec7f40761b6f40a13e1d

A392096

Numbers k for which the width pattern of the symmetric representation of sigma for k, SRS(k), has the form 1 2 1 0 1 2 1 0 . . . 0 1 2 1 0 1 2 1 containing at least one 0.

[ "78", "102", "114", "138", "174", "186", "222", "246", "258", "282", "318", "348", "354", "366", "372", "402", "426", "438", "444", "474", "492", "498", "516", "534", "564", "582", "606", "618", "636", "642", "654", "678", "708", "732", "762", "7...

[ "nonn", "new" ]

12

1

[ "A003056", "A237270", "A237271", "A237593", "A249223", "A249351", "A367377", "A370205", "A377654", "A392096" ]

null

Hartmut F. W. Hoft, Dec 30 2025

2026-01-06T15:29:10

oeisdata/seq/A392/A392096.seq

15e0c76ecb29e3e8ead3fb6aa9aa8ef3

A392097

a(n) is the smallest number k for which the symmetric representation of sigma for k, SRS(k), has n parts and each i-th part up to the diagonal is unimodal of width i.

[ "1", "3", "15", "75", "1225", "1275", "2145", "43125" ]

[ "nonn", "more", "new" ]

9

1

2

[ "A003056", "A237270", "A237271", "A237591", "A237593", "A249223", "A249351", "A377654", "A392097" ]

null

Hartmut F. W. Hoft, Dec 30 2025

2026-01-11T19:21:29

oeisdata/seq/A392/A392097.seq

2c6d82f1547f046adcc982c9e773ddc9

A392098

Powers k^m divisible by at least 2 cubes greater than 1, with m > 1 and nonsquarefree weak k (in A332785).

[ "1728", "5832", "8000", "13824", "20736", "21952", "32400", "63504", "64000", "85184", "90000", "91125", "104976", "110592", "125000", "129600", "140608", "156816", "157464", "160000", "175616", "202500", "216000", "219024", "248832", "250047", "254016", "291600...

[ "nonn", "easy", "new" ]

8

1

[ "A001597", "A001694", "A013929", "A024619", "A062503", "A126706", "A131605", "A286708", "A332785", "A359280", "A386762", "A389864", "A391968", "A392069", "A392098" ]

null

Michael De Vlieger, Jan 08 2026

2026-01-14T17:48:11

oeisdata/seq/A392/A392098.seq

a3f32d0b6a3d9127be016ebb7b40f1dc

A392099

Smallest prime p such that p + 3^n - 1 is also a prime.

[ "2", "3", "3", "3", "3", "29", "5", "17", "3", "5", "3", "71", "17", "17", "3", "3", "29", "89", "11", "137", "47", "167", "5", "107", "3", "29", "3", "17", "71", "131", "5", "17", "71", "71", "59", "23", "3", "197", "179", "83", "233",...

[ "nonn", "new" ]

21

0

1

[ "A000040", "A001359", "A051783", "A056206", "A342502", "A392099" ]

null

James S. DeArmon, Dec 30 2025

2026-01-08T22:20:21

oeisdata/seq/A392/A392099.seq

04a2ee731a3ff13627a8b8320e882dbd

A392100

Array listed by rows: Pythagorean triples a < b < c such that a + b and a + b + 2*c are prime, sorted first by increasing perimeter and then by the even term.

[ "3", "4", "5", "5", "12", "13", "28", "45", "53", "11", "60", "61", "36", "77", "85", "65", "72", "97", "15", "112", "113", "24", "143", "145", "88", "105", "137", "119", "120", "169", "28", "195", "197", "21", "220", "221", "60", "221", "2...

[ "nonn", "tabf", "new" ]

8

1

[ "A103605", "A103606", "A392100" ]

null

Will Gosnell and Robert Israel, Dec 30 2025

2026-01-05T15:16:56

oeisdata/seq/A392/A392100.seq

eefece70248efa3ee267e31813f4ade1

A392101

Numbers whose square is the concatenation of two or more powers of 2.

[ "9", "11", "12", "18", "21", "22", "29", "38", "41", "46", "91", "92", "108", "109", "111", "119", "121", "122", "129", "168", "204", "208", "221", "338", "341", "358", "359", "378", "404", "408", "471", "478", "482", "642", "696", "804", "908"...

[ "nonn", "base", "new" ]

59

1

[ "A272884", "A392101", "A392102" ]

null

Rhys Feltman, Dec 30 2025

2026-01-12T13:59:40

oeisdata/seq/A392/A392101.seq

4ef89783fff664213a24b722f7e26b6e

A392102

Squares which are the concatenation of two or more powers of 2.

[ "81", "121", "144", "324", "441", "484", "841", "1444", "1681", "2116", "8281", "8464", "11664", "11881", "12321", "14161", "14641", "14884", "16641", "28224", "41616", "43264", "48841", "114244", "116281", "128164", "128881", "142884", "163216", "166464", ...

[ "nonn", "base", "new" ]

68

1

[ "A000290", "A272884", "A381259", "A392101", "A392102" ]

null

Rhys Feltman, Dec 30 2025

2026-01-12T16:06:40

oeisdata/seq/A392/A392102.seq

13dfd47c37b33d22f054bd2e33c97d39

A392104

a(n) = prime(A391796(n)).

[ "5", "23", "47", "251", "1889", "7793", "43451", "243161", "726893", "759821", "1820123", "1820111", "10141499", "19725479", "19725473", "136209239", "400414127", "400414121", "489144659", "489144599", "766319201", "766319189", "21549657581", "21549657551", "215496575...

[ "nonn", "new" ]

16

1

[ "A391796", "A391807", "A392104" ]

null

Clark Kimberling, Jan 08 2026

2026-01-17T12:56:33

oeisdata/seq/A392/A392104.seq

c2509ad83b84a5bef72472d1bffdfd17

A392106

Concatenation of binary words using letters 1 and 2, in numerical order.

[ "1", "2", "1", "1", "1", "2", "2", "1", "2", "2", "1", "1", "1", "1", "1", "2", "1", "2", "1", "1", "2", "2", "2", "1", "1", "2", "1", "2", "2", "2", "1", "2", "2", "2", "1", "1", "1", "1", "1", "1", "1", "2", "1", "1", "2", "...

[ "nonn", "easy", "new" ]

11

0

2

[ "A007931", "A076478", "A392106", "A392107" ]

null

Clark Kimberling, Jan 09 2026

2026-01-14T19:23:54

oeisdata/seq/A392/A392106.seq

6ae25c7e1e584b9d157d3e1f0f7ecc88

A392108

a(n) = smallest prime with exponent > 1 in the prime factorization of n, or 0 if no such prime exists.

[ "0", "0", "0", "2", "0", "0", "0", "2", "3", "0", "0", "2", "0", "0", "0", "2", "0", "3", "0", "2", "0", "0", "0", "2", "5", "0", "3", "2", "0", "0", "0", "2", "0", "0", "0", "2", "0", "0", "0", "2", "0", "0", "0", "2", "3", "...

[ "nonn", "changed" ]

19

1

4

[ "A056170", "A063958", "A080368", "A392066", "A392108", "A392109" ]

null

Paolo Xausa, Dec 31 2025

2026-01-07T03:09:20

oeisdata/seq/A392/A392108.seq

319d0dcc2f8b79e79d96ea02c6320075

A392109

a(n) = largest prime with exponent > 1 in the prime factorization of n, or 0 if no such prime exists.

[ "0", "0", "0", "2", "0", "0", "0", "2", "3", "0", "0", "2", "0", "0", "0", "2", "0", "3", "0", "2", "0", "0", "0", "2", "5", "0", "3", "2", "0", "0", "0", "2", "0", "0", "0", "3", "0", "0", "0", "2", "0", "0", "0", "2", "3", "...

[ "nonn", "changed" ]

23

1

4

[ "A027748", "A056170", "A063958", "A080367", "A392066", "A392108", "A392109" ]

null

Paolo Xausa, Dec 31 2025

2026-01-17T23:10:46

oeisdata/seq/A392/A392109.seq

f2de5133abb0b232fe537cffe8f2c239

A392110

Numbers k such that 5*k - (greatest prime < 5*n) < (least prime > 5*n) - 5*k.

[ "4", "5", "11", "15", "16", "17", "18", "22", "23", "28", "35", "37", "40", "43", "46", "47", "49", "53", "57", "59", "64", "68", "70", "71", "72", "75", "76", "77", "78", "82", "85", "87", "88", "89", "90", "94", "96", "100", "101", "102...

[ "nonn", "new" ]

15

1

[ "A000027", "A000040", "A390788", "A392110", "A392111", "A392112" ]

null

Clark Kimberling, Jan 11 2026

2026-01-15T13:08:17

oeisdata/seq/A392/A392110.seq

c2331c40b36c5fbb8b97c73b7e851772

A392111

Numbers k such that 5*k - (greatest prime < 5*n) = (least prime > 5*n) - 5*k.

[ "1", "3", "6", "9", "10", "12", "21", "24", "30", "32", "33", "34", "36", "39", "41", "45", "48", "52", "54", "60", "63", "74", "81", "84", "93", "99", "103", "112", "114", "118", "120", "122", "123", "125", "129", "130", "132", "135", "136...

[ "nonn", "new" ]

8

1

2

[ "A000027", "A000040", "A390788", "A392110", "A392111", "A392112" ]

null

Clark Kimberling, Jan 15 2026

2026-01-18T23:43:47

oeisdata/seq/A392/A392111.seq

87b6466d51065f2862ad06102c55d9f7

A392112

Numbers k such that 5*k - (greatest prime < 5*n) > (least prime > 5*n) - 5*k.

[ "2", "7", "8", "13", "14", "19", "20", "25", "26", "27", "29", "31", "38", "42", "44", "50", "51", "55", "56", "58", "61", "62", "65", "66", "67", "69", "73", "79", "80", "83", "86", "91", "92", "95", "97", "98", "104", "107", "108", "109...

[ "nonn", "new" ]

8

1

[ "A000027", "A000040", "A390788", "A392110", "A392111", "A392112" ]

null

Clark Kimberling, Jan 15 2026

2026-01-18T23:43:23

oeisdata/seq/A392/A392112.seq

f1cae2134574a1cac780239967c70917

A392113

Numbers k such that 6*k - (greatest prime < 6*n) < (least prime > 6*n) - 6*k.

[ "4", "8", "9", "14", "15", "19", "22", "28", "29", "34", "36", "39", "42", "43", "44", "49", "53", "59", "60", "64", "65", "67", "74", "75", "78", "80", "82", "84", "85", "88", "93", "94", "98", "99", "104", "108", "109", "111", "113", "1...

[ "nonn", "new" ]

10

1

[ "A000027", "A000040", "A390788", "A392113", "A392114", "A392115" ]

null

Clark Kimberling, Jan 15 2026

2026-01-18T14:28:54

oeisdata/seq/A392/A392113.seq

48e57bc8cc77931094bd8899654069ae

A392114

Numbers k such that 6*k - (greatest prime < 6*n) = (least prime > 6*n) - 6*k.

[ "1", "2", "3", "5", "7", "10", "12", "17", "18", "20", "23", "24", "25", "30", "31", "32", "33", "38", "40", "41", "45", "47", "48", "50", "52", "54", "57", "58", "69", "70", "71", "72", "77", "87", "92", "95", "97", "100", "103", "106", ...

[ "nonn", "new" ]

6

1

2

[ "A000027", "A000040", "A390788", "A392113", "A392114", "A392115" ]

null

Clark Kimberling, Jan 16 2026

2026-01-18T14:29:18

oeisdata/seq/A392/A392114.seq

3f37103fb6c96ad31ce7f26cbb0d9ad2

A392115

Numbers k such that 6*k - (greatest prime < 6*n) > (least prime > 6*n) - 6*k.

[ "6", "11", "13", "16", "21", "26", "27", "35", "37", "46", "51", "55", "56", "61", "62", "63", "66", "68", "73", "76", "79", "81", "83", "86", "89", "90", "91", "96", "101", "102", "105", "112", "115", "118", "121", "122", "123", "125", "12...

[ "nonn", "new" ]

6

1

[ "A000027", "A000040", "A390788", "A392113", "A392114", "A392115" ]

null

Clark Kimberling, Jan 16 2026

2026-01-18T14:29:43

oeisdata/seq/A392/A392115.seq

50e5077b42ea688f80b5b0293fb68f70

A392126

The smallest k >= 0 that can be represented as a linear combination of 1^3, 2^3, ..., n^3 with coefficients +-1 and that cannot be represented using 1^3, 2^3, ..., m^3 with 1<=m<n.

[ "1", "7", "18", "28", "25", "45", "26", "4", "5", "15", "12", "0", "3", "13", "8", "24", "83", "4463", "6932", "10710", "9269", "27845", "30410", "50660", "47853", "80109", "64610", "117368", "104237", "163709", "139880", "220500", "190709", "289205"...

[ "nonn", "new" ]

19

1

2

[ "A000537", "A000578", "A019568", "A392025", "A392126", "A392127", "A392346" ]

null

Alois P. Heinz, Dec 30 2025

2026-01-08T18:53:12

oeisdata/seq/A392/A392126.seq

355fcb66462060000d57892ec570c65c

A392127

The smallest k >= 0 that can be represented as a linear combination of 1^2, 2^2, ..., n^2 with coefficients +-1 and that cannot be represented using 1^2, 2^2, ..., m^2 with 1<=m<n.

[ "1", "3", "4", "2", "13", "7", "0", "44", "25", "61", "52", "250", "241", "563", "458", "984", "823", "1529", "1272", "2214", "1853", "3055", "2614", "4068", "3539", "5269", "4644", "6674", "5945", "8299", "7458", "10160", "9199", "12273", "11184",...

[ "nonn", "new" ]

21

1

2

[ "A000290", "A000330", "A019568", "A231015", "A392126", "A392127", "A392346" ]

null

Alois P. Heinz, Dec 30 2025

2026-01-08T18:53:06

oeisdata/seq/A392/A392127.seq

56e45a03a105b476421cbea13d97d2ac

A392129

Integers k such that there are i groups of order k+i up to isomorphism, for 1 <= i <= 6.

[ "29436120", "117873720", "222954120", "324510120", "689250312", "876688920", "894332520", "946770120", "1417704120", "1761885720", "1834771320", "2199390120", "2240912520", "2509602120", "2779308072", "3091545720", "3143256120", "3262502520", "3436558920", "3585632472", "3652...

[ "nonn", "new" ]

22

1

[ "A000001", "A003277", "A054395", "A054396", "A054397", "A055561", "A135850", "A249550", "A373648", "A373649", "A373650", "A381335", "A392129", "A392130" ]

null

Gabriel Eiseman, Dec 31 2025

2026-01-06T22:02:43

oeisdata/seq/A392/A392129.seq

a22021b4d11f45d958bbf27312e64268

A392130

Integers k such that there are i groups of order k+1 up to isomorphism, for i=1,2,3,4,5,6,7.

[ "5973822114120", "6305771634120", "28058687347320", "48414128744520", "74556478687320", "84300170172120", "142366076070120", "153432090884520", "207916418382120", "339858901929960", "375278016786120", "415728077892120", "426239950426920" ]

[ "nonn", "more", "new" ]

12

1

[ "A000001", "A003277", "A054395", "A054396", "A054397", "A055561", "A135850", "A249550", "A373648", "A373649", "A373650", "A381335", "A392129", "A392130" ]

null

Gabriel Eiseman, Dec 31 2025

2026-01-12T16:37:11

oeisdata/seq/A392/A392130.seq

a330c792b7a238cbf32a203861d61d46

A392131

a(n) is the determinant of the n X n circulant matrix whose rows are formed by successively rotating the vector (A000129(0), A000129(1), ..., A000129(n-1)) right.

[ "1", "0", "1", "9", "640", "252100", "597791376", "8252767236301", "665993168101048320", "313398861092499186897912", "859858277306561684298748471552", "13751722491910335588624048111158497953", "1281922118247965546737056182821273600000000000", "6965087592028412199949848226918542065689144193...

[ "nonn" ]

10

0

4

[ "A000129", "A086459", "A118713", "A318173", "A392131", "A392132" ]

null

Stefano Spezia, Dec 31 2025

2026-01-02T09:31:36

oeisdata/seq/A392/A392131.seq

6846cafb2b0608c70f3f5e8405a1612f

A392132

a(n) is the permanent of the n X n circulant matrix whose rows are formed by successively rotating the vector (A000129(0), A000129(1), ..., A000129(n-1)) right.

[ "1", "0", "1", "9", "772", "317660", "755353604", "10245351511241", "806176992721352256", "369503021786486764387032", "989096947940706813698038153600", "15475597969523884739005106748390084993", "1415492646595459083986339197781974314319509760", "756726561532204931161254789744755112027328520...

[ "nonn" ]

8

0

4

[ "A000129", "A306457", "A392131", "A392132" ]

null

Stefano Spezia, Dec 31 2025

2026-01-02T09:31:45

oeisdata/seq/A392/A392132.seq

32a26ee5dc176bcba30b155410e72ec1

A392135

Third column of A391957.

[ "0", "0", "0", "0", "0", "9", "7", "5", "3", "1", "18", "14", "10", "6", "2", "27", "21", "15", "9", "3", "36", "28", "20", "12", "4", "94", "82", "70", "58", "46", "93", "79", "65", "51", "37", "92", "76", "60", "44", "28", "91", "73...

[ "nonn", "new" ]

37

0

6

[ "A055457", "A391956", "A391957", "A392135" ]

null

A.H.M. Smeets, Jan 01 2026

2026-01-13T13:40:20

oeisdata/seq/A392/A392135.seq

9758a6c0df3f5cd9bbec89da84f3ecd5

A392136

Fourth column of A391957.

[ "0", "0", "0", "0", "0", "0", "0", "13", "11", "9", "7", "5", "3", "1", "26", "22", "18", "14", "10", "6", "2", "39", "33", "27", "21", "15", "9", "3", "52", "44", "36", "28", "20", "12", "4", "65", "55", "45", "35", "25", "15", "5", ...

[ "nonn", "new" ]

21

0

8

[ "A373217", "A391956", "A391957", "A392136" ]

null

A.H.M. Smeets, Jan 05 2026

2026-01-15T17:53:24

oeisdata/seq/A392/A392136.seq

dda68005ba65f536e8fb0aa6f56b7791

A392137

a(n) = 5*a(n-1) + 2*5^n - 1, a(0) = 0.

[ "0", "9", "94", "719", "4844", "30469", "183594", "1074219", "6152344", "34667969", "192871094", "1062011719", "5798339844", "31433105469", "169372558594", "907897949219", "4844665527344", "25749206542969", "136375427246094", "720024108886719", "3790855407714844", "19907951...

[ "nonn", "easy", "new" ]

21

0

2

[ "A392135", "A392137" ]

null

A.H.M. Smeets, Jan 05 2026

2026-01-15T09:39:05

oeisdata/seq/A392/A392137.seq

4c5e6793c85203a733c081f0fe437f9b

A392138

a(n) = 7*a(n-1) + 2*7^n - 1, a(0) = 0.

[ "0", "13", "188", "2001", "18808", "165269", "1392180", "11392345", "91276016", "719639325", "5602425772", "43171633889", "329884011624", "2502966102181", "18877208860964", "141635585046633", "1057914956465632", "7870665723233837", "58351487258457756", "431258201179950577", "...

[ "nonn", "easy", "new" ]

14

0

2

[ "A392136", "A392138" ]

null

A.H.M. Smeets, Jan 06 2026

2026-01-13T07:58:53

oeisdata/seq/A392/A392138.seq

b7212f73526252213e9aea684a8671a7