Distribution of the number of prime factors with a given multiplicity
Bibliographic record
Abstract
Abstract Given an integer $k\ge 2$ , let $\omega _k(n)$ denote the number of primes that divide n with multiplicity exactly k. We compute the density $e_{k,m}$ of those integers n for which $\omega _k(n)=m$ for every integer $m\ge 0$ . We also show that the generating function $\sum _{m=0}^\infty e_{k,m}z^m$ is an entire function that can be written in the form $\prod _{p} \bigl (1+{(p-1)(z-1)}/{p^{k+1}} \bigr )$ ; from this representation we show how to both numerically calculate the $e_{k,m}$ to high precision and provide an asymptotic upper bound for the $e_{k,m}$ . We further show how to generalize these results to all additive functions of the form $\sum _{j=2}^\infty a_j \omega _j(n)$ ; when $a_j=j-1$ this recovers a classical result of Rényi on the distribution of $\Omega (n)-\omega (n)$ .
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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 | 0.012 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".