On the Powerful and Squarefree Parts of an Integer
Bibliographic record
Abstract
Any integer n ≥ 2 can be written in a unique way as the product of its powerful part and its squarefree part, that is, as n = mr where m is a powerful number and r a squarefree number, with gcd(m,r) = 1. We denote these two parts of an integer n by pow(n) and sq(n) respectively, setting for convenience pow(1) = sq(1) = 1. We first examine the behavior of the counting functions P n≤x,sq(n)≤y 1 and P n≤x,pow(n)≤y 1. Letting P(n) stand for the largest prime factor of n, we then provide asymptotic values of Ay(x) := P n≤x,P(n)≤y pow(n) and By(x) := P n≤x,P(n)≤y sq(n) when y = x 1/u with u ≥ 1 fixed. We also examine the size of Ay(x) and By(x) when y = (logx) η for some � > 1. Finally, we prove that Ay(x) will coincide with By(x) in the sense that log(Ay(x)/x) = (1 + o(1))log(By(x)/x) as x → ∞ if we choose y = 2logx.
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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 | 0.004 | 0.028 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| 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".