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Sums of distinct divisors of factorials
Abstract
For practical N let h(N) be the least k such that every integer 1≤ m≤ N is a sum of at most k distinct divisors of N. We prove h(n!)≤(2log2+o(1)) n/log n. This improves the bounds of order n/(log n)1/2−ε established in Tenenbaum-Yokota's Lemma 4 and Yokota's 1995 knapsack note. We combine their decreasing greedy construction with the sharper factorial divisor-gap estimate of Berend-Harmse. Counting the steps separately below and above √(n!), with the upper range handled through reciprocal divisors, retains the leading coefficient in the gap exponent and yields the explicit constant 2log2.
Related
BibTeX
@misc{hughes2026sums,
title = {{Sums of distinct divisors of factorials}},
author = {Scott D. Hughes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10902},
eprint = {2609.10902},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10902},
}
RIS
TY - UNPB
TI - Sums of distinct divisors of factorials
AU - Scott D. Hughes
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10902
UR - https://arxiv.org/abs/2609.10902
AB - For practical N let h(N) be the least k such that every integer 1≤ m≤ N is a sum of at most k distinct divisors of N. We prove h(n!)≤(2log2+o(1)) n/log n. This improves the bounds of order n/(log n)^1/2−ε established in Tenenbaum-Yokota's Lemma 4 and Yokota's 1995 knapsack note. We combine their decreasing greedy construction with the sharper factorial divisor-gap estimate of Berend-Harmse. Counting the steps separately below and above √(n!), with the upper range handled through reciprocal divisors, retains the leading coefficient in the gap exponent and yields the explicit constant 2log2.
ER -