Datasets:
christopher
/
oeis

Dataset card Data Studio Files
xet
 Community
2
sequence_id
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7
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573
sequence
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1
348
keywords
listlengths
1
7
score
int64
1
2.47k
offset_a
int64
-14,827
666,262,453B
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int64
0
635M
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cross_references
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author
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timestamp
timestamp[us]date
1999-12-11 03:00:00
2026-01-19 02:46:49
filename
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29
29
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stringlengths
32
32
A002001
a(n) = 3*4^(n-1), n>0; a(0)=1.
[ "1", "3", "12", "48", "192", "768", "3072", "12288", "49152", "196608", "786432", "3145728", "12582912", "50331648", "201326592", "805306368", "3221225472", "12884901888", "51539607552", "206158430208", "824633720832", "3298534883328", "13194139533312", "52776558133248"...
[ "nonn", "easy" ]
149
0
2
[ "A000302", "A002001", "A003945", "A006342", "A011782", "A033428", "A134316", "A178789" ]
null
N. J. A. Sloane, Dec 11 1996
2025-02-16T08:32:24
oeisdata/seq/A002/A002001.seq
a367d9eab4656b1957c9c8b64afba0d0
A002002
a(n) = Sum_{k=0..n-1} binomial(n,k+1) * binomial(n+k,k).
[ "0", "1", "5", "25", "129", "681", "3653", "19825", "108545", "598417", "3317445", "18474633", "103274625", "579168825", "3256957317", "18359266785", "103706427393", "586889743905", "3326741166725", "18885056428537", "107347191941249", "610916200215241" ]
[ "nonn", "easy" ]
193
0
3
[ "A001003", "A001850", "A002002", "A002003", "A008288", "A026002", "A047781", "A050143", "A064861", "A190666", "A259554" ]
[ "M3938", "N1621" ]
N. J. A. Sloane, Simon Plouffe
2025-11-05T15:35:24
oeisdata/seq/A002/A002002.seq
a984f3fa9418841b36adf6de5da24e55
A002003
a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k).
[ "0", "2", "8", "38", "192", "1002", "5336", "28814", "157184", "864146", "4780008", "26572086", "148321344", "830764794", "4666890936", "26283115038", "148348809216", "838944980514", "4752575891144", "26964373486406", "153196621856192", "871460014012682", "496289518769704...
[ "nonn" ]
105
0
2
[ "A002002", "A002003", "A006318", "A027307", "A103885", "A108424" ]
[ "M1857", "N0735" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002003.seq
89136cda6a4097a2562a90711276acd7
A002004
Davenport-Schinzel numbers of degree 4 on n symbols.
[ "1", "4", "8", "12", "17", "22", "27", "32", "37", "42", "47", "53", "58", "64", "69", "75", "81", "86", "92", "98", "104" ]
[ "nonn", "nice", "more", "changed" ]
39
1
2
[ "A002004", "A259874" ]
[ "M3328", "N1339" ]
N. J. A. Sloane
2026-01-12T04:19:14
oeisdata/seq/A002/A002004.seq
9efeac469cd4744d70967d33091519b3
A002005
Number of rooted planar cubic maps with 2n vertices.
[ "1", "4", "32", "336", "4096", "54912", "786432", "11824384", "184549376", "2966845440", "48855252992", "820675092480", "14018773254144", "242919827374080", "4261707069259776", "75576645116559360", "1353050213048123392", "24428493151359467520", "444370175232646840320", "8138178...
[ "nonn" ]
100
0
2
[ "A000168", "A000217", "A000260", "A000309", "A000698", "A000699", "A002005", "A062980", "A266240", "A267827", "A358367" ]
[ "M3646", "N1483" ]
N. J. A. Sloane
2025-11-05T15:21:41
oeisdata/seq/A002/A002005.seq
9da4fb38010ac62ab5739ff0f5d826fb
A002006
Almost trivalent maps.
[ "2", "24", "272", "3424", "46720", "676608", "10251520", "160900608" ]
[ "nonn", "more" ]
25
0
1
[ "A002006", "A002007", "A002008", "A002009", "A002010" ]
[ "M2137", "N0849" ]
N. J. A. Sloane
2022-02-04T00:37:37
oeisdata/seq/A002/A002006.seq
d8e12a98cd3e2a4932a7cd432fd756ed
A002007
Almost trivalent maps.
[ "15", "200", "2672", "37600", "554880", "8514560", "134864640" ]
[ "nonn", "more" ]
19
0
1
[ "A002006", "A002007", "A002008", "A002009", "A002010" ]
[ "M4987", "N2144" ]
N. J. A. Sloane
2022-02-04T00:37:29
oeisdata/seq/A002/A002007.seq
40898af04668dc3e590c72c306613c5f
A002008
Almost trivalent maps.
[ "5", "120", "1840", "27552", "421248", "6613504", "106441472", "1750927872" ]
[ "nonn", "more" ]
18
0
1
[ "A002006", "A002007", "A002008", "A002009", "A002010" ]
[ "M4027", "N1671" ]
N. J. A. Sloane
2022-02-04T00:37:20
oeisdata/seq/A002/A002008.seq
7bdba61cbc6d86ebd1eacf75722b65cf
A002009
Almost trivalent maps.
[ "56", "1120", "18592", "300288", "4877824", "80349696", "1344154112" ]
[ "nonn", "more" ]
18
0
1
[ "A002006", "A002007", "A002008", "A002009", "A002010" ]
[ "M5314", "N2308" ]
N. J. A. Sloane
2022-02-04T00:37:12
oeisdata/seq/A002/A002009.seq
6ce01b945c25ba4ee353844b9f227db2
A002010
Almost trivalent maps.
[ "14", "560", "11200", "197568", "3378944", "57573888" ]
[ "nonn", "more" ]
19
0
1
[ "A002006", "A002007", "A002008", "A002009", "A002010" ]
[ "M4955", "N2123" ]
N. J. A. Sloane
2022-02-04T00:37:04
oeisdata/seq/A002/A002010.seq
6f57acd33d6db9524ba2201075329d79
A002011
a(n) = 4*(2n+1)!/n!^2.
[ "4", "24", "120", "560", "2520", "11088", "48048", "205920", "875160", "3695120", "15519504", "64899744", "270415600", "1123264800", "4653525600", "19234572480", "79342611480", "326704870800", "1343120024400", "5513861152800", "22606830726480", "92580354403680", "37873781...
[ "nonn" ]
33
0
1
[ "A001803", "A002011", "A002457", "A005430" ]
[ "M3598", "N1458" ]
N. J. A. Sloane, Simon Plouffe
2025-11-02T03:34:24
oeisdata/seq/A002/A002011.seq
367a5a8ebb207e3c713ff5caea179f79
A002012
Almost trivalent maps.
[ "4", "32", "200", "1120", "5880", "29568", "144144", "686400", "3208920", "14780480", "67251184", "302865472", "1352078000", "5990745600", "26369978400", "115407434880", "502503206040", "2178032472000", "9401840170800", "40434981787200", "173319035569680", "740642835229440"...
[ "nonn" ]
15
0
1
[ "A002005", "A002006", "A002007", "A002008", "A002009", "A002010", "A002011", "A002012" ]
[ "M3643", "N1481" ]
N. J. A. Sloane
2024-06-24T16:46:14
oeisdata/seq/A002/A002012.seq
c3264adb4b90dfecfea9eef7b758ac1f
A002013
Number of filaments with n square cells.
[ "1", "1", "1", "2", "3", "7", "13", "31", "65", "154", "347", "824", "1905", "4512", "10546", "24935", "58476", "138002", "323894", "763172", "1790585", "4213061", "9878541", "23214728", "54393063", "127687369", "298969219", "701171557", "1640683309", "38447...
[ "nonn" ]
56
0
4
[ "A002013", "A003104", "A182644", "A333313" ]
[ "M0835", "N0317" ]
N. J. A. Sloane
2025-12-16T03:29:23
oeisdata/seq/A002/A002013.seq
9d6c771c0c3ae290ef0376f08a856174
A002014
Number of symmetric filaments (strip polyominoes) with n square cells.
[ "1", "1", "1", "2", "2", "5", "5", "12", "12", "27", "28", "64", "67", "147", "158", "348", "373", "799", "879", "1886", "2069", "4335", "4864", "10204", "11432", "23488", "26814", "55179", "62925", "127133", "147414", "298212", "345415" ]
[ "nonn" ]
20
0
4
null
[ "M0349", "N0131" ]
N. J. A. Sloane
2022-02-04T00:36:47
oeisdata/seq/A002/A002014.seq
a56ba5d640a3e1935f1169f009eef333
A002015
a(n) = n^2 reduced mod 100.
[ "0", "1", "4", "9", "16", "25", "36", "49", "64", "81", "0", "21", "44", "69", "96", "25", "56", "89", "24", "61", "0", "41", "84", "29", "76", "25", "76", "29", "84", "41", "0", "61", "24", "89", "56", "25", "96", "69", "44", "21", "0"...
[ "nonn", "easy" ]
27
0
3
[ "A002015", "A053879", "A070430", "A070431", "A070432", "A070433", "A070434", "A070435", "A070438", "A070442", "A070452", "A159852" ]
null
N. J. A. Sloane
2022-02-04T11:02:11
oeisdata/seq/A002/A002015.seq
12578c5321bd487b0b6248407c1074b5
A002016
Number of first n tetrahedral numbers (A000292) that are relatively prime to n.
[ "1", "1", "3", "1", "2", "2", "4", "2", "6", "1", "8", "2", "10", "2", "5", "4", "14", "3", "16", "2", "7", "4", "20", "4", "10", "5", "18", "4", "26", "2", "28", "8", "16", "7", "8", "6", "34", "8", "20", "4", "38", "3", "40", "8...
[ "nonn", "easy" ]
32
1
3
[ "A000292", "A002016" ]
[ "M2212", "N0878" ]
N. J. A. Sloane
2023-07-31T10:08:00
oeisdata/seq/A002/A002016.seq
c3918ce7b9aab7dd73b98a0be2705218
A002017
Expansion of e.g.f. exp(sin(x)).
[ "1", "1", "1", "0", "-3", "-8", "-3", "56", "217", "64", "-2951", "-12672", "5973", "309376", "1237173", "-2917888", "-52635599", "-163782656", "1126610929", "12716052480", "20058390573", "-495644917760", "-3920482183827", "4004259037184", "256734635981833", "135917...
[ "sign", "easy", "nice" ]
76
0
5
[ "A002017", "A003722", "A003724", "A007301", "A047687", "A047688" ]
[ "M2709", "N1086" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002017.seq
bdf7ded9de38655a9e858d35e4374b82
A002018
From a distribution problem.
[ "1", "1", "4", "33", "480", "11010", "367560", "16854390", "1016930880", "78124095000", "7446314383200", "862332613342200", "119261328828364800", "19415283189746043600", "3675162134109650184000", "800409618620667941886000", "198730589981586780813696000", "55800304882692417053710704...
[ "nonn", "easy", "nice" ]
34
0
3
[ "A000681", "A002018" ]
[ "M3652", "N1485" ]
N. J. A. Sloane
2025-04-24T04:32:25
oeisdata/seq/A002/A002018.seq
5c40650883a14f72c71a0711b3e827e9
A002019
a(n) = a(n-1) - (n-1)(n-2)a(n-2).
[ "1", "1", "1", "-1", "-7", "5", "145", "-5", "-6095", "-5815", "433025", "956375", "-46676375", "-172917875", "7108596625", "38579649875", "-1454225641375", "-10713341611375", "384836032842625", "3663118565923375", "-127950804666254375", "-1519935859717136875" ]
[ "sign", "nice", "easy" ]
71
0
5
[ "A000246", "A002019", "A006228", "A049218", "A102058", "A102059" ]
[ "M4330", "N1813" ]
N. J. A. Sloane
2025-11-26T15:59:31
oeisdata/seq/A002/A002019.seq
d112c5d6468359dff79424cff2d0ba3f
A002020
a(n+1) = a(n) - n*(n-1)*a(n-1), with a(n) = 1 for n <= 3.
[ "1", "1", "1", "1", "-5", "-17", "83", "593", "-2893", "-36101", "172195", "3421285", "-15520165", "-467129785", "1954015955", "86971636825", "-323371713725", "-21196564551725", "66760541581475", "6552909294409325", "-16279195926455125", "-2506384727801998625", "433087756...
[ "sign", "easy" ]
17
0
5
null
[ "M3876", "N1588" ]
N. J. A. Sloane
2022-02-04T00:35:41
oeisdata/seq/A002/A002020.seq
2d4b385ed9c1c67138431f70a6c876c5
A002021
Pile of coconuts problem: (n-1)*(n^n - 1), n even; n^n - n + 1, n odd.
[ "1", "3", "25", "765", "3121", "233275", "823537", "117440505", "387420481", "89999999991", "285311670601", "98077104930805", "302875106592241", "144456088732254195", "437893890380859361", "276701161105643274225", "827240261886336764161", "668888937280041138782191", "197841965566...
[ "easy", "nonn", "nice" ]
59
1
2
[ "A002021", "A002022", "A006091" ]
[ "M3114", "N1262" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002021.seq
a64a4a3c4e8bc94a6d2071ee78b34078
A002022
In the pile of coconuts problem, the number of coconuts that remain to be shared equally at the end of the process.
[ "0", "6", "240", "1020", "78120", "279930", "40353600", "134217720", "31381059600", "99999999990", "34522712143920", "106993205379060", "51185893014090744", "155568095557812210", "98526125335693359360", "295147905179352825840", "239072435685151324847136" ]
[ "nonn", "easy", "nice" ]
37
2
2
[ "A002021", "A002022", "A006091" ]
[ "M4305", "N1800" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002022.seq
74fd432c6914a1e31f0a14a2a774988e
A002023
a(n) = 6*4^n.
[ "6", "24", "96", "384", "1536", "6144", "24576", "98304", "393216", "1572864", "6291456", "25165824", "100663296", "402653184", "1610612736", "6442450944", "25769803776", "103079215104", "412316860416", "1649267441664", "6597069766656", "26388279066624", "105553116266496"...
[ "nonn", "easy" ]
88
0
1
[ "A002023", "A004171", "A283070" ]
null
N. J. A. Sloane
2025-02-16T08:32:24
oeisdata/seq/A002/A002023.seq
6d7cd476dc2ab428671a4f7c54640dbf
A002024
k appears k times; a(n) = floor(sqrt(2n) + 1/2).
[ "1", "2", "2", "3", "3", "3", "4", "4", "4", "4", "5", "5", "5", "5", "5", "6", "6", "6", "6", "6", "6", "7", "7", "7", "7", "7", "7", "7", "8", "8", "8", "8", "8", "8", "8", "8", "9", "9", "9", "9", "9", "9", "9", "9", "9", "...
[ "nonn", "easy", "nice", "tabl" ]
312
1
2
[ "A000194", "A001462", "A001563", "A002024", "A002260", "A002262", "A003056", "A003881", "A004736", "A005145", "A014132", "A022846", "A025581", "A060432", "A093995", "A107985", "A123578", "A127899", "A131507" ]
[ "M0250", "N0089" ]
N. J. A. Sloane
2025-11-26T15:59:32
oeisdata/seq/A002/A002024.seq
30f8e4df19c25dae312522b6bae4a1e7
A002025
Smaller of an amicable pair: (a,b) such that sigma(a) = sigma(b) = a+b, a < b.
[ "220", "1184", "2620", "5020", "6232", "10744", "12285", "17296", "63020", "66928", "67095", "69615", "79750", "100485", "122265", "122368", "141664", "142310", "171856", "176272", "185368", "196724", "280540", "308620", "319550", "356408", "437456", "469028", ...
[ "nonn", "nice" ]
115
1
1
[ "A000203", "A002025", "A002046", "A063990", "A066873", "A180164", "A259180" ]
[ "M5414", "N2352" ]
N. J. A. Sloane
2025-07-02T11:34:49
oeisdata/seq/A002/A002025.seq
f67bffdc96c93f3890bf5b6b77a47899
A002026
Generalized ballot numbers (first differences of Motzkin numbers).
[ "0", "1", "2", "5", "12", "30", "76", "196", "512", "1353", "3610", "9713", "26324", "71799", "196938", "542895", "1503312", "4179603", "11662902", "32652735", "91695540", "258215664", "728997192", "2062967382", "5850674704", "16626415975", "47337954326", "13501...
[ "nonn", "easy", "nice" ]
146
0
3
[ "A001006", "A002026", "A020474", "A026107", "A026300", "A244884", "A348840", "A348869" ]
[ "M1416", "N0554" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002026.seq
da14a3415bf516df3024f29d44b2aa89
A002027
Number of connected graphs on n labeled nodes, each node being colored with one of 2 colors, such that no edge joins nodes of the same color.
[ "1", "2", "2", "6", "38", "390", "6062", "134526", "4172198", "178449270", "10508108222", "853219059726", "95965963939958", "15015789392011590", "3282145108526132942", "1005193051984479922206", "432437051675617901246918", "261774334771663762228012950", "22330643752633365772628327...
[ "nonn" ]
28
0
2
[ "A002027", "A002031", "A002032", "A322279" ]
[ "M0365", "N0138" ]
N. J. A. Sloane
2018-12-03T18:27:35
oeisdata/seq/A002/A002027.seq
8c4d72b5fa827f9377bee1e0671cf703
A002028
Number of connected graphs on n labeled nodes, each node being colored with one of 3 colors, such that no edge joins nodes of the same color.
[ "1", "3", "6", "42", "618", "15990", "668526", "43558242", "4373213298", "677307561630", "162826875512646", "61183069270120842", "36134310487980825258", "33673533885068169649830", "49646105434209446798290206", "116002075479856331220877149042", "430053223599741677879550609246498", "...
[ "nonn" ]
35
0
2
[ "A002028", "A002031", "A002032", "A322279" ]
[ "M2603", "N1030" ]
N. J. A. Sloane
2018-12-03T17:46:21
oeisdata/seq/A002/A002028.seq
3342e9234dd3f80218c1a65dc8b6b02f
A002029
Number of connected graphs on n labeled nodes, each node being colored with one of 4 colors, such that no edge joins nodes of the same color.
[ "1", "4", "12", "132", "3156", "136980", "10015092", "1199364852", "234207001236", "75018740661780", "39745330657406772", "35073541377640231092", "51798833078501480220756", "128412490016744675540378580", "535348496386845235339961362932", "3757366291145650829115977555259252" ]
[ "nonn" ]
36
0
2
[ "A002029", "A002032", "A322279" ]
[ "M3459", "N1406" ]
N. J. A. Sloane
2019-09-05T00:04:46
oeisdata/seq/A002/A002029.seq
b48a5ec8ed4e76ac838a73432585653a
A002030
Number of connected graphs on n labeled nodes, each node being colored with one of 5 colors, such that no edge joins nodes of the same color.
[ "1", "5", "20", "300", "9980", "616260", "65814020", "11878194300", "3621432947180", "1880516646144660", "1678121372919602420", "2590609089652498130700", "6947580541943715645962780", "32448510765823652400410879460", "264301377639329321236008592510820" ]
[ "nonn" ]
37
0
2
[ "A002030", "A002032", "A322279" ]
[ "M3911", "N1606" ]
N. J. A. Sloane
2019-09-04T10:23:51
oeisdata/seq/A002/A002030.seq
f86e1ee694c46d9af4dcf38b92561d0b
A002031
Number of labeled connected digraphs on n nodes where every node has indegree 0 or outdegree 0 and no isolated nodes.
[ "2", "6", "38", "390", "6062", "134526", "4172198", "178449270", "10508108222", "853219059726", "95965963939958", "15015789392011590", "3282145108526132942", "1005193051984479922206", "432437051675617901246918", "261774334771663762228012950", "223306437526333657726283273822" ]
[ "nonn", "nice" ]
52
2
1
[ "A001831", "A001832", "A002027", "A002031", "A002032", "A007776", "A047863", "A052332" ]
[ "M1707", "N0676" ]
N. J. A. Sloane
2023-11-18T14:17:39
oeisdata/seq/A002/A002031.seq
b8026206881f61a0701845952e5cec56
A002032
Number of n-colored connected graphs on n labeled nodes.
[ "1", "1", "2", "24", "912", "87360", "19226880", "9405930240", "10142439229440", "24057598104207360", "125180857812868300800", "1422700916050060841779200", "35136968950395142864227532800", "1876028272361273394915958613606400", "215474119792145796020405035320528076800" ]
[ "nonn" ]
42
0
3
[ "A001187", "A002027", "A002028", "A002029", "A002030", "A002031", "A002032", "A322278", "A322279", "A322280" ]
[ "M2141", "N0852" ]
N. J. A. Sloane
2024-01-05T23:32:36
oeisdata/seq/A002/A002032.seq
0bd04c55d4023c9221d9f8d1b067c09a
A002033
Number of perfect partitions of n.
[ "1", "1", "1", "2", "1", "3", "1", "4", "2", "3", "1", "8", "1", "3", "3", "8", "1", "8", "1", "8", "3", "3", "1", "20", "2", "3", "4", "8", "1", "13", "1", "16", "3", "3", "3", "26", "1", "3", "3", "20", "1", "13", "1", "8", "8...
[ "nonn", "core", "easy", "nice" ]
122
0
4
[ "A000123", "A000670", "A001055", "A001678", "A002033", "A002110", "A003238", "A008966", "A050324", "A067824", "A074206", "A100529", "A108917", "A117621", "A126796", "A167865", "A176917", "A214577", "A289078", "A292504", "A316782" ]
[ "M0131", "N0053" ]
N. J. A. Sloane
2025-10-23T19:00:37
oeisdata/seq/A002/A002033.seq
635e71ad0fda78bf3ee2ee37a5a76d9c
A002034
Kempner numbers: smallest positive integer m such that n divides m!.
[ "1", "2", "3", "4", "5", "3", "7", "4", "6", "5", "11", "4", "13", "7", "5", "6", "17", "6", "19", "5", "7", "11", "23", "4", "10", "13", "9", "7", "29", "5", "31", "8", "11", "17", "7", "6", "37", "19", "13", "5", "41", "7", "43", ...
[ "nonn", "nice", "easy" ]
241
1
2
[ "A000142", "A001113", "A002034", "A006530", "A007672", "A046022", "A050376", "A057109", "A064759", "A084945", "A094371", "A094372", "A094404", "A122378", "A122379", "A122416", "A122417", "A248937", "A339594", "A339596" ]
[ "M0453", "N0167" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002034.seq
ae21464cf5bbd1650c2e7c2df6ffc532
A002035
Numbers that contain primes to odd powers only.
[ "2", "3", "5", "6", "7", "8", "10", "11", "13", "14", "15", "17", "19", "21", "22", "23", "24", "26", "27", "29", "30", "31", "32", "33", "34", "35", "37", "38", "39", "40", "41", "42", "43", "46", "47", "51", "53", "54", "55", "56", "5...
[ "nonn", "nice" ]
52
1
1
[ "A000203", "A002035", "A036537", "A065463", "A072586", "A072587", "A124010", "A162644", "A188999", "A295316" ]
[ "M0614", "N0224" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002035.seq
a732e9453291960ad459b10e3f6f922a
A002036
Compressed primes: a(n) is the nearest integer to prime(n)/log prime(n).
[ "3", "3", "3", "4", "5", "5", "6", "6", "7", "9", "9", "10", "11", "11", "12", "13", "14", "15", "16", "17", "17", "18", "19", "20", "21", "22", "22", "23", "23", "24", "26", "27", "28", "28", "30", "30", "31", "32", "33", "34", "35", ...
[ "nonn" ]
28
1
1
[ "A002036", "A070944" ]
[ "M2272", "N0898" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002036.seq
8c5d4c6cb128434053beb28e16ce7225
A002037
Product of all primes up to 3^n.
[ "1", "6", "210", "223092870", "3217644767340672907899084554130", "256041159035492609053110100510385311995538591998443060216114576417920917800321526504084465112487730" ]
[ "nonn" ]
31
0
2
null
[ "M4303", "N1799" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002037.seq
3927056f0d82826ec344e450c2c1d6e3
A002038
Numbers dividing A002037(i) and larger than A002037(i-1), for some i>0.
[ "1", "2", "3", "6", "7", "10", "14", "15", "21", "30", "35", "42", "70", "105", "210", "221", "230", "231", "238", "247", "253", "255", "266", "273", "285", "286", "299", "322", "323", "330", "345", "357", "374", "385", "390", "391", "399", "...
[ "nonn" ]
30
1
2
[ "A002037", "A002038" ]
[ "M0749", "N0282" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002038.seq
8a97a6310e9d2049605ee2e429fa9850
A002039
Convolution inverse of A143348.
[ "1", "3", "5", "10", "25", "64", "160", "390", "940", "2270", "5515", "13440", "32735", "79610", "193480", "470306", "1143585", "2781070", "6762990", "16445100", "39987325", "97232450", "236432060", "574915770", "1397981470", "3399360474", "8265943685", "2009961...
[ "nonn", "nice", "easy" ]
42
0
2
[ "A002039", "A002040", "A143348" ]
[ "M2465", "N0979" ]
N. J. A. Sloane, Simon Plouffe
2025-11-05T15:35:24
oeisdata/seq/A002/A002039.seq
a6f0ce7ee0aa5b391f951369687f5d8b
A002040
Related to partitions.
[ "1", "2", "4", "8", "21", "52", "131", "316", "765", "1846", "4494", "10944", "26654", "64798", "157502", "382868", "931028", "2264106", "5505777", "13387880", "32553601", "79156974", "192479838", "468039888", "1138098210", "2767421826", "6729311459", "163631185...
[ "nonn", "easy", "nice" ]
44
0
2
[ "A000203", "A002039", "A002040", "A010815" ]
[ "M1159", "N0442" ]
N. J. A. Sloane
2025-07-08T16:27:32
oeisdata/seq/A002/A002040.seq
5461be00e8b074a8991f242f751750e6
A002041
Expansion of x/((1-x)(1-4x^2)(1-5x)).
[ "1", "6", "35", "180", "921", "4626", "23215", "116160", "581141", "2906046", "14531595", "72659340", "363302161", "1816516266", "9082603175", "45413037720", "227065275981", "1135326467286", "5676632685955", "28383163779300", "141915820294601", "709579102871106", "3547895...
[ "nonn", "easy" ]
46
1
2
null
[ "M4216", "N1759" ]
N. J. A. Sloane
2022-08-27T19:00:51
oeisdata/seq/A002/A002041.seq
772171a29dc1fe94d95375441da3b1ef
A002042
a(n) = 7*4^n.
[ "7", "28", "112", "448", "1792", "7168", "28672", "114688", "458752", "1835008", "7340032", "29360128", "117440512", "469762048", "1879048192", "7516192768", "30064771072", "120259084288", "481036337152", "1924145348608", "7696581394432", "30786325577728", "12314530231091...
[ "nonn", "easy" ]
79
0
1
[ "A000290", "A000302", "A000578", "A002042", "A005009", "A009971", "A083597", "A306472" ]
null
N. J. A. Sloane
2025-09-22T16:00:16
oeisdata/seq/A002/A002042.seq
7d54b9ca42c49f1254d116c8ffc79a89
A002043
MacMahon's solid partitions of n in which 2 is the smallest summand.
[ "3", "0", "6", "18", "40", "81", "201", "414", "916", "1899", "3973", "8059", "16402", "32561", "64520", "125986", "244448", "469195", "895077", "1692143", "3179406", "5929721", "10993373", "20250589", "37096872", "67568512", "122437970", "220721343", "3959988...
[ "nonn" ]
27
2
1
[ "A000294", "A002043" ]
[ "M2204", "N1710" ]
N. J. A. Sloane
2026-01-04T14:15:01
oeisdata/seq/A002/A002043.seq
e54dc1f05a1c4581876cefa4b347ae18
A002044
MacMahon's solid partitions of n in which 3 is the smallest summand.
[ "6", "0", "0", "21", "60", "90", "182", "378", "861", "1737", "3458", "6717", "13377", "25877", "49949", "95085", "180254", "338003", "631124", "1168226", "2151409", "3934674", "7159108", "12948649", "23307439", "41730764", "74374385", "131927754", "233011531"...
[ "nonn", "changed" ]
30
3
1
[ "A000294", "A002043", "A002044", "A002045" ]
[ "M4041", "N2214" ]
N. J. A. Sloane
2026-01-05T09:36:49
oeisdata/seq/A002/A002044.seq
1207d50c908761d0f1fd3d768728245b
A002045
MacMahon's solid partitions of n in which 4 is the smallest summand.
[ "10", "0", "0", "0", "55", "150", "210", "280", "580", "1275", "2905", "5350", "9985", "17965", "33665", "62895", "117287", "214610", "389805", "700720", "1259890", "2250405", "4008717", "7092366", "12497237", "21904825", "38253450", "66511772", "115230973", ...
[ "nonn", "changed" ]
26
4
1
[ "A000294", "A002043", "A002044", "A002045" ]
[ "M4674", "N2307" ]
N. J. A. Sloane
2026-01-05T09:36:41
oeisdata/seq/A002/A002045.seq
e359543821406fbcd60a91e69c326079
A002046
Larger of amicable pair.
[ "284", "1210", "2924", "5564", "6368", "10856", "14595", "18416", "76084", "66992", "71145", "87633", "88730", "124155", "139815", "123152", "153176", "168730", "176336", "180848", "203432", "202444", "365084", "389924", "430402", "399592", "455344", "486178", ...
[ "nonn", "nice" ]
97
1
1
[ "A002025", "A002046", "A063990", "A180164", "A259180" ]
[ "M5435", "N2363" ]
N. J. A. Sloane
2025-07-02T11:34:45
oeisdata/seq/A002/A002046.seq
ad9c7a7cf4e6bee0d950989b13ae6745
A002047
Number of 3 X (2n+1) zero-sum arrays with entries -n,...,0,...,n.
[ "1", "2", "6", "28", "244", "2544", "35600", "659632", "15106128", "425802176", "14409526080", "577386122880", "27029459961728", "1462450710036480" ]
[ "nonn", "nice", "more" ]
97
0
2
[ "A002047", "A014552", "A260333", "A309260", "A309746" ]
[ "M1688", "N0666" ]
N. J. A. Sloane
2025-12-07T16:00:58
oeisdata/seq/A002/A002047.seq
43376720e53dbb7effeab9d35af50b3d
A002048
Segmented numbers, or prime numbers of measurement.
[ "1", "2", "4", "5", "8", "10", "14", "15", "16", "21", "22", "25", "26", "28", "33", "34", "35", "36", "38", "40", "42", "46", "48", "49", "50", "53", "57", "60", "62", "64", "65", "70", "77", "80", "81", "83", "85", "86", "90", "91", "...
[ "nonn", "nice" ]
93
1
2
[ "A002048", "A002049", "A004978", "A005242", "A033627" ]
[ "M0972", "N0363" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002048.seq
24892386bb15df746228350f711bf8c2
A002049
Prime numbers of measurement.
[ "1", "3", "7", "12", "20", "30", "44", "59", "75", "96", "118", "143", "169", "197", "230", "264", "299", "335", "373", "413", "455", "501", "549", "598", "648", "701", "758", "818", "880", "944", "1009", "1079", "1156", "1236", "1317", "1400", ...
[ "nonn", "nice" ]
46
1
2
[ "A002048", "A002049", "A004978", "A048204" ]
[ "M2633", "N1044" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002049.seq
2dbd4a7b6b2e4b2b740c8b35a4747967
A002050
Number of simplices in barycentric subdivision of n-simplex.
[ "0", "1", "5", "25", "149", "1081", "9365", "94585", "1091669", "14174521", "204495125", "3245265145", "56183135189", "1053716696761", "21282685940885", "460566381955705", "10631309363962709", "260741534058271801", "6771069326513690645" ]
[ "nonn", "easy", "nice" ]
94
0
3
[ "A000629", "A000670", "A002050", "A052856", "A053440", "A076726", "A241168" ]
[ "M3939", "N1622" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002050.seq
af6e2a9e91f133c5c05a71fa0be70c5a
A002051
Steffensen's bracket function [n,2].
[ "0", "0", "1", "9", "67", "525", "4651", "47229", "545707", "7087005", "102247051", "1622631549", "28091565547", "526858344285", "10641342962251", "230283190961469", "5315654681948587", "130370767029070365", "3385534663256714251", "92801587319328148989", "26776877962443836788...
[ "nonn", "easy", "nice" ]
61
1
4
[ "A000670", "A002051", "A011782", "A019472", "A033312", "A056823", "A056986", "A241168", "A333217", "A335447", "A335454", "A335456", "A335465", "A335485", "A335486", "A335515" ]
[ "M4644", "N1986" ]
N. J. A. Sloane
2020-06-24T19:05:01
oeisdata/seq/A002/A002051.seq
0aa64c3a7a4a3402d77341c60b92b5d9
A002052
Prime determinants of forms with class number 2.
[ "3", "7", "11", "19", "23", "31", "43", "47", "59", "67", "71", "83", "103", "107", "127", "131", "139", "151", "163", "167", "179", "191", "199", "211", "227", "239", "251", "263", "271", "283", "307", "311", "331", "347", "367", "379", "383",...
[ "nonn" ]
34
1
1
[ "A002052", "A002145", "A260335" ]
[ "M4339", "N1816" ]
N. J. A. Sloane
2025-11-24T10:12:47
oeisdata/seq/A002/A002052.seq
1f366a37d2dc7e0e73712c4df6139c81
A002053
a(n) = least value of m for which Liouville's function A002819(m) = -n.
[ "2", "3", "8", "13", "20", "31", "32", "53", "76", "79", "80", "117", "176", "181", "182", "193", "200", "283", "284", "285", "286", "293", "440", "443", "468", "661", "678", "683", "684", "1075", "1076", "1087", "1088", "1091", "1092", "1093", ...
[ "nonn", "nice", "look" ]
33
0
1
[ "A002053", "A008836", "A028488", "A072203" ]
[ "M0871", "N0333" ]
N. J. A. Sloane
2023-01-02T02:17:12
oeisdata/seq/A002/A002053.seq
05e50a001bfd525452d0186653587989
A002054
Binomial coefficient C(2n+1, n-1).
[ "1", "5", "21", "84", "330", "1287", "5005", "19448", "75582", "293930", "1144066", "4457400", "17383860", "67863915", "265182525", "1037158320", "4059928950", "15905368710", "62359143990", "244662670200", "960566918220", "3773655750150", "14833897694226", "583433568174...
[ "nonn", "easy" ]
277
1
2
[ "A000097", "A000346", "A000984", "A001263", "A001622", "A001700", "A001791", "A002054", "A002694", "A002696", "A003516", "A004310", "A004318", "A007318", "A008549", "A024483", "A030053", "A030056", "A031444", "A031445", "A033282", "A037951", "A037955", "A058622", "A09...
[ "M3913", "N1607" ]
N. J. A. Sloane
2025-12-22T11:35:18
oeisdata/seq/A002/A002054.seq
58016cfdd968314f912ff86a5a118aab
A002055
Number of diagonal dissections of a convex n-gon into n-4 regions.
[ "1", "9", "56", "300", "1485", "7007", "32032", "143208", "629850", "2735810", "11767536", "50220040", "212952285", "898198875", "3771484800", "15775723920", "65770848990", "273420862110", "1133802618000", "4691140763400", "19371432850770", "79850555673174", "328627887379...
[ "nonn", "nice", "easy" ]
65
5
2
[ "A001622", "A002055", "A033282", "A034261" ]
[ "M4639", "N1982" ]
N. J. A. Sloane
2025-12-22T11:35:24
oeisdata/seq/A002/A002055.seq
fc534ae1ce1e145e52e059b165d13f34
A002056
Number of diagonal dissections of a convex n-gon into n-5 regions.
[ "1", "14", "120", "825", "5005", "28028", "148512", "755820", "3730650", "17978180", "84987760", "395482815", "1816357725", "8250123000", "37119350400", "165645101160", "733919156190", "3231337461300", "14147884842000", "61636377252450", "267325773340626", "1154761882042824...
[ "nonn", "easy" ]
82
6
2
[ "A001622", "A002056", "A033282" ]
[ "M4941", "N2115" ]
N. J. A. Sloane
2025-12-22T11:35:21
oeisdata/seq/A002/A002056.seq
6a0bb3014eb2e5da977cce62e5aa5a70
A002057
Fourth convolution of Catalan numbers: a(n) = 4*binomial(2*n+3,n)/(n+4).
[ "1", "4", "14", "48", "165", "572", "2002", "7072", "25194", "90440", "326876", "1188640", "4345965", "15967980", "58929450", "218349120", "811985790", "3029594040", "11338026180", "42550029600", "160094486370", "603784920024", "2282138106804", "8643460269248", "32798...
[ "nonn", "easy", "nice" ]
282
0
2
[ "A000108", "A000245", "A000344", "A000588", "A001003", "A001392", "A001622", "A002057", "A003517", "A003518", "A003519", "A009766", "A030237", "A033184", "A047072", "A059365", "A099039", "A106566", "A130020", "A145596", "A279004" ]
[ "M3483", "N1415" ]
N. J. A. Sloane
2025-11-26T15:59:32
oeisdata/seq/A002/A002057.seq
de7c4c30cab5f784c12342ecfc87fad4
A002058
Number of internal triangles in all triangulations of an (n+1)-gon.
[ "2", "14", "72", "330", "1430", "6006", "24752", "100776", "406980", "1634380", "6537520", "26075790", "103791870", "412506150", "1637618400", "6495886320", "25751549340", "102042235620", "404225281200", "1600944863700", "6339741660252", "25103519174844", "99399793096352"...
[ "nonn", "easy" ]
44
5
1
[ "A001622", "A002058", "A002059", "A002060", "A003516" ]
[ "M2069", "N0817" ]
N. J. A. Sloane
2025-10-24T06:21:36
oeisdata/seq/A002/A002058.seq
e4524a68cf5a48308e2d4c438822c041
A002059
Number of partitions of an n-gon into (n-4) parts.
[ "3", "32", "225", "1320", "7007", "34944", "167076", "775200", "3517470", "15690048", "69052555", "300638520", "1297398375", "5557977600", "23663585880", "100222246080", "422559514170", "1774647576000", "7427639542050", "30994292561232", "128989359164358", "535541742396672"...
[ "nonn", "easy" ]
45
6
1
[ "A001622", "A002058", "A002059", "A002060" ]
[ "M3130", "N1269" ]
N. J. A. Sloane
2025-10-24T06:21:13
oeisdata/seq/A002/A002059.seq
7cdfc0fed812c0298269ee5f8d2dcccb
A002060
Number of partitions of an n-gon into (n-5) parts.
[ "4", "60", "550", "4004", "25480", "148512", "813960", "4263600", "21573816", "106234700", "511801290", "2421810300", "11289642000", "51967090560", "236635858800", "1067518772640", "4776759725400", "21221827263000", "93687293423724", "411270420524040", "1796296260955504", "...
[ "nonn" ]
28
7
1
[ "A002058", "A002059", "A002060" ]
[ "M3691", "N1509" ]
N. J. A. Sloane
2023-03-10T03:18:38
oeisdata/seq/A002/A002060.seq
9d0986e3a2d85af7e56f6ecfaf7ff16d
A002061
Central polygonal numbers: a(n) = n^2 - n + 1.
[ "1", "1", "3", "7", "13", "21", "31", "43", "57", "73", "91", "111", "133", "157", "183", "211", "241", "273", "307", "343", "381", "421", "463", "507", "553", "601", "651", "703", "757", "813", "871", "931", "993", "1057", "1123", "1191", "126...
[ "nonn", "easy", "nice" ]
591
0
3
[ "A000037", "A000124", "A000217", "A000290", "A000384", "A001107", "A001263", "A001844", "A002061", "A002378", "A002383", "A002620", "A002939", "A002943", "A004273", "A005408", "A005563", "A007645", "A007742", "A010000", "A014105", "A014206", "A016742", "A016754", "A02...
[ "M2638", "N1049" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002061.seq
2a69636a0ff1496a6d3ddb67da6d6be4
A002062
a(n) = Fibonacci(n) + n.
[ "0", "2", "3", "5", "7", "10", "14", "20", "29", "43", "65", "100", "156", "246", "391", "625", "1003", "1614", "2602", "4200", "6785", "10967", "17733", "28680", "46392", "75050", "121419", "196445", "317839", "514258", "832070", "1346300", "2178341",...
[ "nonn", "easy", "nice" ]
94
0
2
[ "A000045", "A001611", "A002062", "A160536", "A212272" ]
[ "M0646", "N0240" ]
N. J. A. Sloane
2025-09-22T16:00:16
oeisdata/seq/A002/A002062.seq
bfcebdebe4cc4bfffb61e06c55d96f98
A002063
a(n) = 9*4^n.
[ "9", "36", "144", "576", "2304", "9216", "36864", "147456", "589824", "2359296", "9437184", "37748736", "150994944", "603979776", "2415919104", "9663676416", "38654705664", "154618822656", "618475290624", "2473901162496", "9895604649984", "39582418599936", "15832967439974...
[ "nonn", "easy" ]
99
0
1
[ "A000302", "A002001", "A002063", "A055841" ]
null
N. J. A. Sloane
2025-04-10T02:47:05
oeisdata/seq/A002/A002063.seq
80d43a7166e95c77de1d41a09edf8db1
A002064
Cullen numbers: a(n) = n*2^n + 1.
[ "1", "3", "9", "25", "65", "161", "385", "897", "2049", "4609", "10241", "22529", "49153", "106497", "229377", "491521", "1048577", "2228225", "4718593", "9961473", "20971521", "44040193", "92274689", "192937985", "402653185", "838860801", "1744830465", "3623878...
[ "nonn", "easy", "nice" ]
198
0
2
[ "A000005", "A000169", "A000272", "A000312", "A000325", "A001787", "A002064", "A002109", "A003261", "A005849", "A036289", "A046688", "A050914", "A057156", "A062319", "A066959", "A130197", "A134081", "A143038", "A156708", "A173339", "A176029", "A181527", "A186947", "A18...
[ "M2795", "N1125" ]
N. J. A. Sloane
2025-11-05T15:21:41
oeisdata/seq/A002/A002064.seq
25945ad1b4c8d5e87cff1bbc2b65a6d4
A002065
a(n+1) = a(n)^2 + a(n) + 1.
[ "0", "1", "3", "13", "183", "33673", "1133904603", "1285739649838492213", "1653126447166808570252515315100129583", "2732827050322355127169206170438813672515557678636778921646668538491883473" ]
[ "easy", "nice", "nonn" ]
112
0
3
[ "A002065", "A002665", "A002794", "A002795", "A030125", "A063573", "A232806" ]
[ "M2961", "N1197" ]
N. J. A. Sloane
2025-11-26T15:59:32
oeisdata/seq/A002/A002065.seq
00a270bc71c7f45d77e9436fc212b560
A002066
a(n) = 10*4^n.
[ "10", "40", "160", "640", "2560", "10240", "40960", "163840", "655360", "2621440", "10485760", "41943040", "167772160", "671088640", "2684354560", "10737418240", "42949672960", "171798691840", "687194767360", "2748779069440", "10995116277760", "43980465111040", "175921860...
[ "nonn", "easy" ]
42
0
1
[ "A000079", "A002066", "A003947", "A004171", "A020714", "A081294" ]
null
N. J. A. Sloane
2025-04-02T15:16:56
oeisdata/seq/A002/A002066.seq
7d16c14e661a987602da5bc493c432fc
A002067
a(n) = Sum_{k=0..n-1} binomial(2*n,2*k)*a(k)*a(n-k-1).
[ "1", "1", "7", "127", "4369", "243649", "20036983", "2280356863", "343141433761", "65967241200001", "15773461423793767", "4591227123230945407", "1598351733247609852849", "655782249799531714375489", "313160404864973852338669783", "172201668512657346455126457343", "10802634947676204112...
[ "nonn", "eigen", "easy", "nice" ]
103
0
3
[ "A002067", "A092676", "A122149", "A122159", "A132467" ]
[ "M4458", "N1889" ]
N. J. A. Sloane
2025-11-05T15:21:41
oeisdata/seq/A002/A002067.seq
01088781f827be6c1ebf9733b4fd6af9
A002068
Wilson remainders: a(n) = ((p-1)!+1)/p mod p, where p = prime(n).
[ "1", "1", "0", "5", "1", "0", "5", "2", "8", "18", "19", "7", "16", "13", "6", "34", "27", "56", "12", "69", "11", "73", "20", "70", "70", "72", "57", "1", "30", "95", "71", "119", "56", "67", "94", "86", "151", "108", "21", "106", "48"...
[ "nonn", "nice", "easy" ]
52
1
4
[ "A002068", "A007540", "A007619", "A275741" ]
[ "M3728", "N1524" ]
N. J. A. Sloane
2025-11-05T15:21:41
oeisdata/seq/A002/A002068.seq
e0e421074319c0b4161fc6b2fb6dff13
A002069
Palindromic pentagonal numbers.
[ "0", "1", "5", "22", "1001", "2882", "15251", "720027", "7081807", "7451547", "26811862", "54177145", "1050660501", "1085885801", "1528888251", "2911771192", "2376574756732", "5792526252975", "5875432345785", "10810300301801", "264571020175462", "5292834004382925", "10808...
[ "nonn", "base" ]
34
1
3
[ "A000326", "A002069", "A028386" ]
[ "M3924", "N1612" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002069.seq
e036bca534fbc0e5ae5451a823f2a5af
A002070
Coefficient of x^p (p = n-th prime) in x * Product_{k>=1} (1-x^k)^2*(1-x^11k)^2.
[ "-2", "-1", "1", "-2", "1", "4", "-2", "0", "-1", "0", "7", "3", "-8", "-6", "8", "-6", "5", "12", "-7", "-3", "4", "-10", "-6", "15", "-7", "2", "-16", "18", "10", "9", "8", "-18", "-7", "10", "-10", "2", "-7", "4", "-12", "-6", "-15",...
[ "sign", "easy", "nice" ]
55
1
1
[ "A002070", "A006571", "A006962" ]
[ "M0072", "N0024" ]
N. J. A. Sloane, Sep 13 2003
2023-10-09T12:50:31
oeisdata/seq/A002/A002070.seq
dc7b3aa108e55b601039bdd89303a652
A002071
Number of pairs of consecutive integers x, x+1 such that all prime factors of both x and x+1 are at most the n-th prime.
[ "1", "4", "10", "23", "40", "68", "108", "167", "241", "345", "482", "653", "869", "1153", "1502", "1930", "2454", "3106", "3896", "4839", "6040" ]
[ "nonn", "nice", "hard", "more" ]
89
1
2
[ "A002071", "A002072", "A085152", "A085153", "A138180", "A145604", "A145605", "A145606", "A285283" ]
[ "M3386", "N1366" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002071.seq
08db080441f587dabb635c0ad11eb5d1
A002072
a(n) = smallest number m such that for all k > m, either k or k+1 has a prime factor > prime(n).
[ "1", "8", "80", "4374", "9800", "123200", "336140", "11859210", "11859210", "177182720", "1611308699", "3463199999", "63927525375", "421138799639", "1109496723125", "1453579866024", "20628591204480", "31887350832896", "31887350832896", "119089041053696", "2286831727304144", ...
[ "nonn", "nice", "hard" ]
81
1
2
[ "A002071", "A002072", "A003032", "A003033", "A117581", "A122463", "A145606", "A175607" ]
[ "M4560", "N1942" ]
N. J. A. Sloane
2025-11-05T15:35:24
oeisdata/seq/A002/A002072.seq
1b7433d73377bac26f68bd197e0af165
A002073
Numerators of coefficients in an asymptotic expansion of the confluent hypergeometric function F(1-b; 2; 4b).
[ "1", "-3", "3", "2", "-48", "-362", "-49711", "13952", "574406627", "64140842", "-841796802304", "-326397876886", "-23544490420768844", "45123679545344", "449339765798227104271", "17766371321955738181048", "-20395677580116057792512", "-74026374065532274752108118" ]
[ "sign" ]
26
0
2
[ "A002073", "A002074" ]
[ "M2268", "N0897" ]
N. J. A. Sloane
2023-10-14T15:39:17
oeisdata/seq/A002/A002073.seq
06c88c90589d4ecb9e2ab569fd4cbe10
A002074
Denominators of coefficients in an asymptotic expansion of the confluent hypergeometric function F(1-b; 2; 4b).
[ "1", "5", "35", "225", "67375", "853125", "955040625", "1861234375", "151365980390625", "142468185234375", "10686017754521484375", "8684623124912109375", "5398544111530990341796875", "54231540104196533203125", "1161721704933873029968505859375" ]
[ "nonn", "frac" ]
24
0
2
[ "A002073", "A002074" ]
[ "M3976", "N1645" ]
N. J. A. Sloane
2023-10-14T15:39:54
oeisdata/seq/A002/A002074.seq
5c3cdd71980cdd0a1a0132d1566e056a
A002075
Number of equivalence classes with primitive period n of base 3 necklaces, where necklaces are equivalent under rotation and permutation of symbols.
[ "1", "1", "2", "4", "8", "22", "52", "140", "366", "992", "2684", "7404", "20440", "56992", "159440", "448540", "1266080", "3587610", "10195276", "29057520", "83018728", "237737984", "682196948", "1961323740", "5648590728", "16294032160", "47071589778", "1361714...
[ "nonn", "easy", "nice" ]
18
1
3
[ "A000013", "A000048", "A002075", "A002076" ]
[ "M1160", "N0443" ]
N. J. A. Sloane
2022-01-29T01:16:28
oeisdata/seq/A002/A002075.seq
c52628e06b3c6e8642ec92a6256ebbec
A002076
Number of equivalence classes of base-3 necklaces of length n, where necklaces are considered equivalent under both rotations and permutations of the symbols.
[ "1", "1", "2", "3", "6", "9", "26", "53", "146", "369", "1002", "2685", "7434", "20441", "57046", "159451", "448686", "1266081", "3588002", "10195277", "29058526", "83018783", "237740670", "682196949", "1961331314", "5648590737", "16294052602", "47071590147", ...
[ "nonn", "easy", "nice" ]
66
0
3
[ "A000013", "A000048", "A002075", "A002076", "A056353", "A182522", "A320743" ]
[ "M0761", "N0288" ]
N. J. A. Sloane
2019-05-25T19:29:56
oeisdata/seq/A002/A002076.seq
040a91b8d9a0ff04d4ebc8e935020d61
A002077
Number of N-equivalence classes of self-dual threshold functions of exactly n variables.
[ "1", "0", "1", "4", "46", "1322", "112519", "32267168", "34153652752" ]
[ "nonn", "more" ]
38
1
4
[ "A002077", "A002078", "A002079", "A002080" ]
[ "M3683", "N1503" ]
N. J. A. Sloane
2025-07-03T10:55:37
oeisdata/seq/A002/A002077.seq
bcb519babd8040052ca78fbfc3cdef01
A002078
N-equivalence classes of threshold functions of n or fewer variables.
[ "2", "3", "6", "20", "150", "3287", "244158", "66291591", "68863243522" ]
[ "nonn", "more" ]
48
0
1
[ "A000609", "A002077", "A002078", "A002079", "A002080", "A075271" ]
[ "M0816", "N0308" ]
N. J. A. Sloane
2025-06-28T17:45:28
oeisdata/seq/A002/A002078.seq
f4e977d65a2a2da2b1b0decf61ee261d
A002079
Number of N-equivalence classes of threshold functions of exactly n variables.
[ "2", "1", "2", "9", "96", "2690", "226360", "64646855", "68339572672" ]
[ "nonn", "more" ]
31
0
1
[ "A002077", "A002078", "A002079", "A002080" ]
[ "M0122", "N0049" ]
N. J. A. Sloane
2025-06-27T21:17:32
oeisdata/seq/A002/A002079.seq
55dca90a64718b4dc685671fff030e63
A002080
Number of N-equivalence classes of self-dual threshold functions of n or fewer variables.
[ "1", "2", "4", "12", "81", "1684", "122921", "33207256", "34448225389" ]
[ "nonn", "more" ]
48
1
2
[ "A000609", "A002077", "A002078", "A002080" ]
[ "M1266", "N0485" ]
N. J. A. Sloane
2025-06-27T21:18:02
oeisdata/seq/A002/A002080.seq
4c09ca1e0a832f51fd526b84dff8a418
A002081
Numbers congruent to {2, 4, 8, 16} (mod 20).
[ "2", "4", "8", "16", "22", "24", "28", "36", "42", "44", "48", "56", "62", "64", "68", "76", "82", "84", "88", "96", "102", "104", "108", "116", "122", "124", "128", "136", "142", "144", "148", "156", "162", "164", "168", "176", "182", "184",...
[ "nonn", "easy", "nice" ]
65
0
1
[ "A002081", "A002082", "A008587" ]
[ "M1113", "N0426" ]
N. J. A. Sloane
2023-12-14T06:17:01
oeisdata/seq/A002/A002081.seq
04c9ca253d77f69ab433d96ed0ee14bd
A002082
2nd differences are periodic.
[ "2", "2", "4", "10", "16", "28", "48", "76", "110", "144", "182", "222", "264", "310", "356", "408", "468", "536", "610", "684", "762", "842", "924", "1010", "1096", "1188", "1288", "1396", "1510", "1624", "1742", "1862", "1984", "2110", "2236", ...
[ "nonn", "easy" ]
15
0
1
[ "A002081", "A002082" ]
[ "M0335", "N0127" ]
N. J. A. Sloane
2023-10-14T21:08:08
oeisdata/seq/A002/A002082.seq
4860e9ae03e57b96efdea070756d297a
A002083
Narayana-Zidek-Capell numbers: a(n) = 1 for n <= 2. Otherwise a(2n) = 2a(2n-1), a(2n+1) = 2a(2n) - a(n).
[ "1", "1", "1", "2", "3", "6", "11", "22", "42", "84", "165", "330", "654", "1308", "2605", "5210", "10398", "20796", "41550", "83100", "166116", "332232", "664299", "1328598", "2656866", "5313732", "10626810", "21253620", "42505932", "85011864", "170021123...
[ "easy", "core", "nonn", "nice" ]
159
1
4
[ "A001045", "A002083", "A002487", "A045690", "A058222", "A101688", "A167948", "A242729", "A245094", "A259858" ]
[ "M0787", "N0297" ]
N. J. A. Sloane
2025-08-11T05:58:43
oeisdata/seq/A002/A002083.seq
12244fe4e98ea91fa26bdfe5f54edc65
A002084
Sinh(x) / cos(x) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.
[ "1", "4", "36", "624", "18256", "814144", "51475776", "4381112064", "482962852096", "66942218896384", "11394877025289216", "2336793875186479104", "568240131312188379136", "161669933656307658932224", "53204153193639888357113856", "20053432927718528320240287744" ]
[ "nonn", "easy" ]
52
0
2
[ "A002084", "A002085" ]
[ "M3667", "N1493" ]
N. J. A. Sloane
2025-11-10T21:50:37
oeisdata/seq/A002/A002084.seq
3e44559609f81639189a49976dc4ad98
A002085
From the expansion of sinh(x) / cos(x): a(n) = odd part of A002084(n).
[ "1", "1", "9", "39", "1141", "12721", "804309", "17113719", "1886573641", "65373260641", "11127809595009", "570506317184199", "138730500808639741", "9867549661639871761", "3247323803322747092109", "305991103023048833011479", "130958280255560469436519441", "1578992907627749134268656...
[ "nonn", "easy" ]
37
0
3
[ "A002084", "A002085" ]
[ "M4620", "N1973" ]
N. J. A. Sloane
2025-07-22T10:31:50
oeisdata/seq/A002/A002085.seq
1471ec3c79d5a36d4945241ce239b09f
A002086
Number of circulant tournaments on 2n+1 nodes up to Cayley isomorphism.
[ "1", "1", "2", "4", "4", "6", "16", "16", "30", "88", "94", "208", "472", "586", "1096", "3280", "5472", "7286", "21856", "26216", "49940", "175104", "182362", "399480", "1048576", "1290556", "3355456", "7456600", "9256396", "17895736", "59660288", "8947...
[ "nonn" ]
38
1
3
[ "A002086", "A002087", "A049288" ]
[ "M0939", "N0353" ]
N. J. A. Sloane
2023-10-14T21:11:08
oeisdata/seq/A002/A002086.seq
398f6d54a5bf818bcbfe7d581b768651
A002087
Number of point-symmetric tournaments with 2n+1 nodes.
[ "1", "1", "2", "3", "4", "6", "16", "16", "30" ]
[ "nonn", "more" ]
28
1
3
[ "A002086", "A002087", "A049288" ]
[ "M0578", "N0211" ]
N. J. A. Sloane
2023-10-14T21:11:41
oeisdata/seq/A002/A002087.seq
105ab3e9a640e2b9f92c0780db539ac8
A002088
Sum of totient function: a(n) = Sum_{k=1..n} phi(k), cf. A000010.
[ "0", "1", "2", "4", "6", "10", "12", "18", "22", "28", "32", "42", "46", "58", "64", "72", "80", "96", "102", "120", "128", "140", "150", "172", "180", "200", "212", "230", "242", "270", "278", "308", "324", "344", "360", "384", "396", "432",...
[ "nonn", "easy", "nice" ]
228
0
3
[ "A000010", "A000217", "A001088", "A002088", "A002321", "A005728", "A008683", "A015614", "A018805", "A020652", "A024916", "A038567", "A063985", "A067282", "A098198", "A134542", "A214803" ]
[ "M1008", "N0376" ]
N. J. A. Sloane
2025-12-29T03:53:06
oeisdata/seq/A002/A002088.seq
ce6d6df13d06a36a068d5094bc7eea82
A002089
a(n) = 11*4^n.
[ "11", "44", "176", "704", "2816", "11264", "45056", "180224", "720896", "2883584", "11534336", "46137344", "184549376", "738197504", "2952790016", "11811160064", "47244640256", "188978561024", "755914244096", "3023656976384", "12094627905536", "48378511622144", "193514046...
[ "nonn", "easy" ]
48
0
1
[ "A002089", "A005015", "A072261" ]
null
N. J. A. Sloane
2025-04-02T14:29:32
oeisdata/seq/A002/A002089.seq
e1759a6a00ec7a22b1d14b7447e07d89
A002090
Related to Hamilton numbers.
[ "3", "4", "6", "11", "45", "906", "409182", "83762797735" ]
[ "nonn" ]
12
1
1
null
[ "M2327", "N0920" ]
N. J. A. Sloane
2023-10-14T21:12:11
oeisdata/seq/A002/A002090.seq
9fc2c1f913bf88d7b09ddf6c0a96ba67
A002091
From a Goldbach conjecture: the location of records in A185091.
[ "3", "9", "19", "21", "55", "115", "193", "323", "611", "1081", "1571", "10771", "13067", "16321", "44881", "57887", "93167", "189947", "404939", "442307", "1746551", "3383593", "3544391", "5056787", "7480667", "25619213", "87170987", "404940757", "526805663",...
[ "nonn" ]
43
1
1
[ "A002091", "A002092", "A185091", "A194828", "A194829" ]
[ "M2773", "N1116" ]
N. J. A. Sloane
2023-10-14T21:15:13
oeisdata/seq/A002/A002091.seq
12d37b9ce8df779f61791aab55722e0e
A002092
From a Goldbach conjecture: records in A185091.
[ "1", "3", "5", "7", "17", "29", "47", "61", "73", "83", "277", "317", "349", "419", "503", "601", "709", "829", "877", "1129", "1237", "1367", "1429", "1669", "1801", "2467", "2833", "2879", "3001", "3037", "3329", "3821", "4861", "5003", "5281", ...
[ "nonn" ]
45
1
2
[ "A002091", "A002092", "A002372", "A002373", "A002374", "A002375", "A006307", "A045917", "A185091", "A194828", "A194829" ]
[ "M2424", "N0959" ]
N. J. A. Sloane
2023-10-14T23:45:24
oeisdata/seq/A002/A002092.seq
2b5679ba1d3164d0040adfae30b0dd38
A002093
Highly abundant numbers: numbers k such that sigma(k) > sigma(m) for all m < k.
[ "1", "2", "3", "4", "6", "8", "10", "12", "16", "18", "20", "24", "30", "36", "42", "48", "60", "72", "84", "90", "96", "108", "120", "144", "168", "180", "210", "216", "240", "288", "300", "336", "360", "420", "480", "504", "540", "600", "...
[ "nonn", "nice" ]
86
1
2
[ "A000203", "A002093", "A004394", "A005153", "A034091", "A034885", "A193988", "A193989" ]
[ "M0553", "N0200" ]
N. J. A. Sloane
2022-05-14T11:58:15
oeisdata/seq/A002/A002093.seq
54c5a010089384540c638023daa47398
A002094
Number of unlabeled connected loop-less graphs on n nodes containing exactly one cycle (of length at least 2) and with all nodes of degree <= 4.
[ "0", "1", "2", "5", "10", "25", "56", "139", "338", "852", "2145", "5513", "14196", "36962", "96641", "254279", "671640", "1781840", "4742295", "12662282", "33898923", "90981264", "244720490", "659591378", "1781048728", "4817420360", "13050525328", "35405239155"...
[ "nonn" ]
75
1
3
[ "A000294", "A000598", "A000602", "A000625", "A000642", "A001429", "A002094", "A068051" ]
[ "M1383", "N0541" ]
N. J. A. Sloane
2022-06-18T23:02:57
oeisdata/seq/A002/A002094.seq
6edbe15c150fa7f96372b9c678634db8
A002095
Number of partitions of n into nonprime parts.
[ "1", "1", "1", "1", "2", "2", "3", "3", "5", "6", "8", "8", "12", "13", "17", "19", "26", "28", "37", "40", "52", "58", "73", "79", "102", "113", "139", "154", "191", "210", "258", "284", "345", "384", "462", "509", "614", "679", "805", "...
[ "nonn", "easy", "nice", "changed" ]
64
0
5
[ "A000607", "A002095", "A005171", "A018252", "A023895", "A096258", "A222656" ]
[ "M0271", "N0094" ]
N. J. A. Sloane
2026-01-12T21:26:42
oeisdata/seq/A002/A002095.seq
5c6076147bc89c955ed2f02734fe3e75
A002096
Mixed partitions of n.
[ "0", "0", "1", "2", "3", "6", "9", "14", "20", "29", "42", "58", "79", "108", "145", "191", "252", "329", "427", "549", "704", "894", "1136", "1427", "1793", "2237", "2789", "3450", "4268", "5248", "6447", "7880", "9619", "11691", "14199", "17166...
[ "nonn" ]
17
1
4
null
[ "M0757", "N0286" ]
N. J. A. Sloane
2016-11-08T20:52:18
oeisdata/seq/A002/A002096.seq
fd5176ecefc9f00fc20a0431a73d2e1e
A002097
Numbers that are not the sum of 3 nonzero triangular numbers.
[ "1", "2", "4", "6", "11", "20", "29" ]
[ "fini", "full", "nonn" ]
15
1
2
[ "A002097", "A064825", "A111638" ]
null
N. J. A. Sloane, Dan Hoey
2017-10-06T17:16:22
oeisdata/seq/A002/A002097.seq
4f55f0f24ce746b0b3acd8ade77dc65f
A002098
G.f.: 1/Product_{k>=1} (1-prime(k)*x^prime(k)).
[ "1", "0", "2", "3", "4", "11", "17", "29", "49", "85", "144", "226", "404", "603", "1025", "1679", "2558", "4201", "6677", "10190", "16599", "25681", "39643", "61830", "96771", "147114", "228338", "352725", "533291", "818624", "1263259", "1885918", "29...
[ "nonn" ]
44
0
3
[ "A002098", "A002099", "A006906", "A064364", "A116864" ]
[ "M0597", "N0215" ]
N. J. A. Sloane
2022-02-27T10:23:11
oeisdata/seq/A002/A002098.seq
f14ef22ec1d98867507319b2b3506752
A002099
G.f.: -1 + Product_{k>=1} (1 + prime(k)*x^prime(k)).
[ "0", "0", "2", "3", "0", "11", "0", "17", "15", "14", "51", "11", "77", "35", "103", "131", "160", "227", "330", "218", "426", "794", "426", "1533", "641", "2072", "2071", "2491", "4381", "3880", "3925", "6079", "5446", "7914", "14017", "10421", ...
[ "nonn" ]
17
0
3
[ "A002098", "A002099" ]
[ "M0395", "N0152" ]
N. J. A. Sloane
2023-10-14T15:50:38
oeisdata/seq/A002/A002099.seq
2c03263bfcf88d845600d6b4a18b0f98
A002100
a(n) = number of partitions of n into semiprimes (more precisely, number of ways of writing n as a sum of products of 2 distinct primes).
[ "0", "0", "0", "0", "0", "1", "0", "0", "0", "1", "0", "1", "0", "1", "1", "1", "0", "1", "0", "2", "2", "2", "0", "2", "1", "3", "2", "3", "1", "4", "2", "4", "3", "5", "4", "7", "3", "6", "5", "8", "6", "10", "6", "10", "9", ...
[ "nonn" ]
26
1
20
[ "A002100", "A006881", "A073576", "A101048" ]
[ "M0205", "N0076" ]
N. J. A. Sloane
2020-11-26T12:19:28
oeisdata/seq/A002/A002100.seq
4cc722eb33d97ce021f66849bd436f0f