Three Cousins of Recamán’s Sequence
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
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.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
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 it