The number of large prime factors of integers and normal numbers
Bibliographic record
Abstract
In a series of papers, we constructed large families of normal numbers using the concatenation of the values of the largest prime factor P ( n ) , as n runs through particular sequences of positive integers. A similar approach using the smallest prime factor function also allowed for the construction of normal numbers. Letting ω ( n ) stand for the number of distinct prime factors of the positive integer n , we then showed that the concatenation of the successive values of | ω ( n ) - ⌊ log log n ⌋ | in a fixed base q ≥ 2 , as n runs through the integers n ≥ 3 , yields a normal number. Here we prove the following. Let q ≥ 2 be a fixed integer. Given an integer n ≥ n 0 = max ( q , 3 ) , let N be the unique positive integer satisfying q N ≤ n < q N + 1 and let h ( n , q ) stand for the residue modulo q of the number of distinct prime factors of n located in the interval [ log N , N ] . Setting x N : = e N , we then create a normal number in base q using the concatenation of the numbers h ( n , q ) , as n runs through the integers ≥ x n 0 .
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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 | 0.002 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.008 | 0.002 |
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".