Three Cousins of Recamán’s Sequence
Bibliographic record
Abstract
Although 10230 terms of Recamán’s sequence have been computed, it remains a mystery. Here three distant cousins of that sequence are described, one of which is also mysterious. (i) {A(n), n ≥ 3} is defined as follows. Start with n, and add n + 1, n + 2, n + 3, . . ., stopping after adding n + k if the sum n + (n + 1) + . . . is a multiplicative analog of {A(n)}. Start with n, and successively multiply by n + 1, n + 2, . . ., stopping after multiplying by n+k if the product n(n+1)⋯(n+k) is divisible by n+k+1. Then B(n) = k. We conjecture that log 2 B(n) = ( 1/2 + o(1)) log n log log n. (iii) The third sequence, {C(n), n ≥ 1}, is the most interesting, because it is the most mysterious. Concatenate the decimal digits of n, n + 1, n + 2, . . . until the concatenation n∥n + 1∥ . . . ∥n + k is divisible by n + k + 1. Then C(n) = k. If no such k exists, we set C(n) = −1. We have found k for all n ≤ 1000 except for two cases. Some of the numbers involved are quite large. For example, C(92) = 218128159460, and the concatenation 92∥93∥ . . . ∥(92+C(92))is a number with about 2.1012 digits. We have only a probabilistic argument that such a k exists for all n.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.012 | 0.004 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".