On the Powerful and Squarefree Parts of an Integer
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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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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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