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S. D. Hughes

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Preprint Sep 2026

The density of sums of distinct divisors

For a positive integer $t$, let $d_t$ denote the natural density of the set of $n$ for which $t$ is a sum of distinct divisors of $n$. Erd\H{o}s proved that $d_t$ exists, gave an unspecified polylogarithmic upper bound, asserted without proof a matching lower bound, and asked whether $d_t \sim c_3/(\log t)^{c_4}$. We r...

S. D. Hughes · 0 citations

Preprint Sep 2026

Sums of distinct divisors of factorials

For practical $N$ let $h(N)$ be the least $k$ such that every integer $1\le m\le N$ is a sum of at most $k$ distinct divisors of $N$. We prove $h(n!)\le(2\log2+o(1))\,n/\log n$. This improves the bounds of order $n/(\log n)^{1/2-\varepsilon}$ established in Tenenbaum-Yokota's Lemma 4 and Yokota's 1995 knapsack note. We...

S. D. Hughes · 0 citations