The density of sums of distinct divisors
Abstract
For a positive integer t, let dt denote the natural density of the set of n for which t is a sum of distinct divisors of n. Erdős proved that dt exists, gave an unspecified polylogarithmic upper bound, asserted without proof a matching lower bound, and asked whether dt ∼ c3/(log t)c4. We record the explicit bounds νt ≤ dt ≪ ((loglog t)δ−3/2)/((log t)δ), νt = K/(log t)(1 + O((loglog t)/(log t))), where δ= 0.086071… is the Erdős--Ford--Tenenbaum constant, K=c e−γ, and c=1.33607… is the practical-number constant. Consequently, if Erdős's asymptotic holds, then δ<c4≤ 1. A two-prime construction, using a half-scale sumset to obtain full residue coverage, then yields the pointwise excess liminft→∞(log t) (dt−νt) ≥ KI, where I=∫02 G(w) dw=0.05887… is an explicit elementary integral. In particular dt≥ 0.79/log t for every sufficiently large t, and dt is not asymptotic to K/log t.
Related
BibTeX
@misc{hughes2026density,
title = {{The density of sums of distinct divisors}},
author = {Scott D. Hughes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25446},
eprint = {2609.25446},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25446},
}
RIS
TY - UNPB
TI - The density of sums of distinct divisors
AU - Scott D. Hughes
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25446
UR - https://arxiv.org/abs/2609.25446
AB - 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. Erdős proved that d_t exists, gave an unspecified polylogarithmic upper bound, asserted without proof a matching lower bound, and asked whether d_t ∼ c_3/(log t)^c_4. We record the explicit bounds ν_t ≤ d_t ≪ ((loglog t)^δ−3/2)/((log t)^δ), ν_t = K/(log t)(1 + O((loglog t)/(log t))), where δ= 0.086071… is the Erdős--Ford--Tenenbaum constant, K=c e^−γ, and c=1.33607… is the practical-number constant. Consequently, if Erdős's asymptotic holds, then δ<c_4≤ 1. A two-prime construction, using a half-scale sumset to obtain full residue coverage, then yields the pointwise excess liminf_t→∞(log t) (d_t−ν_t) ≥ KI, where I=∫_0^2 G(w) dw=0.05887… is an explicit elementary integral. In particular d_t≥ 0.79/log t for every sufficiently large t, and d_t is not asymptotic to K/log t.
ER -