N8Programs/oeis-ladder-10b

OEIS Ladder 10B

A 10-billion-token pretraining corpus of integer sequences, built from the On-Line Encyclopedia of Integer Sequences (OEIS): every training-split OEIS sequence once, plus 9.2B tokens of distinct, transformed copies of OEIS sequences. It was used to pretrain NextTerm-2, a scaling ladder of byte-level language models (5M, 50M and 440M parameters) for a study of in-distribution vs out-of-distribution generalization in post-training.

Each row is one sequence written as comma-separated integers, e.g. {"id": "A000045_copy0", "text": "0,1,1,2,3,5,8,13,21,34,..."}.

Composition

	Rows	Tokens*	Share
Real OEIS sequences (A……_copy0)	370,869	754,629,901	7.5%
Generated, transformed sequences (A……_gen…)	12,544,402	9,245,370,098	92.5%
Train total	12,915,271	9,999,999,999
Validation (held-out OEIS sequences)	19,035	3.7M

*Tokens are byte-level training targets: the UTF-8 bytes of text plus one end-of-sequence token per row.

Rows are shuffled. The train split is in data/train-000NN-of-00010.jsonl (about 1 GB each); the held-out set is data/validation.jsonl.

How it was built

Real rows. The term list of each training-split OEIS sequence, taken from its b-file and truncated to a prefix of at most 4,096 bytes. Each appears exactly once.

Generated rows. Repeatedly:

1. Draw a source sequence from the real rows, with probability proportional to 1/√(its length in tokens). Uniform sampling spends most of the budget on very long sequences; 1/length over-samples short, idiosyncratic ones.
2. Transform it (aug_0):
   • with probability 0.3 (sequences of ≥6 terms), one length-reducing transform: take every 2nd or 3rd term, first differences, or a prefix of ≥⅔ of the terms;
   • then 0–2 length-preserving transforms, each chosen uniformly from: multiply by 2–5, negate, negate every other term, add a constant in [−10, 10], swap adjacent pairs, partial sums, reduce mod 2–10.
3. Cap at 4,096 bytes, and require at least 3 terms.
4. Reject exact duplicates of any real row or any earlier generated row, and draw again. 48.3% of draws were rejected; every generated row is distinct.

Generation statistics (sources drawn, rejection counts, seed) are in generation_manifest.json.

Validation split and overlap

The validation sequences are a random 5% of OEIS A-numbers, held out of the real rows and never used as augmentation sources, with exact duplicates of training sequences removed. They double as the OEIS-Eval-Neo benchmark (the 19,034 rows with at least two terms): predict the last term from the preceding ones.

OEIS is full of near-duplicates and transform relationships between entries (e.g. every-other-term of n⁵ is (2n)⁵), so held-out does not mean unseen. For 586 validation sequences (3.1%), some training row starts with the complete validation sequence: 102 via real OEIS entries that extend it, the rest via generated rows (only 11 are exact duplicates). Models score higher on those rows, but excluding them lowers OEIS-Eval-Neo by only 0.6–0.9 points for the NextTerm-2 models (1.0 for NextTerm-440M).

Models trained on it

The NextTerm-2 models: Qwen3-shaped byte-level models (vocabulary of 256 bytes + BOS/EOS/PAD), one pass over the train split, Muon + AdamW, cosine schedule, with identical hyperparameters at every size (Muon learning rate 2e-3, AdamW β2 0.99):

Model	Val loss (nats/byte)	OEIS-Eval-Neo	M1 macro MAPE	Ryskina & Knight
NextTerm-2-5M	1.134	23.4%	20.85	24/57
NextTerm-2-50M	0.960	31.6%	18.36	38/57
NextTerm-2-440M	0.881	36.3%	17.11	38/57

For reference, the M1 competition's Naive2 baseline scores 17.80 macro MAPE, and NextTerm-440M (trained on 14B tokens of a 1/length-sampled predecessor of this recipe) scores 34.4% on OEIS-Eval-Neo.

Loading

from datasets import load_dataset

train = load_dataset("N8Programs/oeis-ladder-10b", split="train", streaming=True)
val = load_dataset("N8Programs/oeis-ladder-10b", split="validation")

License and attribution

The sequences come from The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, by The OEIS Foundation Inc., and are used under the Creative Commons Attribution-ShareAlike 4.0 license. The generated rows are adaptations of OEIS sequences and are shared under the same license.