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arXiv · 2308.07570

On the counts of p-rough numbers

Fred B. Holt
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Abstract

The p-rough numbers are those numbers all of whose prime factors are greater than p. These are exactly those numbers left after Eratosthenes sieve has been advanced from 2 through the prime p. Here we show that for fixed p there is a line of symmetry for the function $\Phi(x,p)$, and we introduce the function $\Delta \Phi(x,p)$ which is the difference between $\Phi(x,p)$ and the line of symmetry. $\Delta \Phi(x,p)$ is periodic and bounded and has a rotational symmetry.

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Fred B. Holt. 2023-08-15. On the counts of p-rough numbers. https://arxiv.org/abs/2308.07570

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