On the parity of the number of multiplicative partitions and related problems
Bibliographic record
Abstract
Let $f(N)$ be the number of unordered factorizations of $N$, where a factorization is a way of writing $N$ as a product of integers all larger than $1$. For example, the factorizations of $30$ are \[ 2\cdot 3\cdot 5,\quad 5\cdot 6, \quad 3\cdot 10, \quad 2 \cdot 15,\quad 30, \] so that $f(30)=5$. The function $f(N)$, as a multiplicative analogue of the (additive) partition function $p(N)$, was first proposed by MacMahon, and its study was pursued by Oppenheim, Szekeres and Turán, and others. Recently, Zaharescu and Zaki showed that $f(N)$ is even a positive proportion of the time and odd a positive proportion of the time. Here we show that for any arithmetic progression $a\operatorname {mod} m$, the set of $N$ for which \[ f(N) \equiv a( \operatorname {mod} m) \] possesses an asymptotic density. Moreover, the density is positive as long as there is at least one such $N$. For the case investigated by Zaharescu and Zaki, we show that $f$ is odd more than 50 percent of the time (in fact, about 57 percent).
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How this classification was reachedexpand
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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".