On the digits of the sum of proper divisors
Abstract
We study several probabilistic questions concerning the digits of s(n), the sum of proper divisors of an integer n. In particular, we show that s(n) obeys Benford's law with respect to logarithmic density. Moreover, we show that, for every function k(x) → ∞, almost all integers n ≤ x have every decimal digit occurring among the first k(x) digits and the last k(x) digits of s(n). We also present an upper bound for the number of composite integers n up to x for which s(n) is missing at least one digit in its decimal expansion. This is in contrast with the main result of a recent paper of Benli, Cesana, Dartyge, Dombrowsky, and Thompson, in which the inputs n were not required to be composite. It turns out that the primes make a substantial contribution to the preimage set s−1(𝒜), where 𝒜 is a set of integers with missing digits. Our result for composite n shows that the count is much smaller when prime inputs are excluded.
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BibTeX
@misc{benli2026digits,
title = {{On the digits of the sum of proper divisors}},
author = {Kübra Benli and Cécile Dartyge and Charlotte Dombrowsky and Paul Pollack and Lola Thompson},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.18981},
eprint = {2607.18981},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18981},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the digits of the sum of proper divisors
AU - Kübra Benli
AU - Cécile Dartyge
AU - Charlotte Dombrowsky
AU - Paul Pollack
AU - Lola Thompson
PY - 2026
DA - 2026/07/21
PB - arXiv
DO - 10.48550/arxiv.2607.18981
UR - https://arxiv.org/abs/2607.18981
AB - We study several probabilistic questions concerning the digits of s(n), the sum of proper divisors of an integer n. In particular, we show that s(n) obeys Benford's law with respect to logarithmic density. Moreover, we show that, for every function k(x) → ∞, almost all integers n ≤ x have every decimal digit occurring among the first k(x) digits and the last k(x) digits of s(n). We also present an upper bound for the number of composite integers n up to x for which s(n) is missing at least one digit in its decimal expansion. This is in contrast with the main result of a recent paper of Benli, Cesana, Dartyge, Dombrowsky, and Thompson, in which the inputs n were not required to be composite. It turns out that the primes make a substantial contribution to the preimage set s^−1(𝒜), where 𝒜 is a set of integers with missing digits. Our result for composite n shows that the count is much smaller when prime inputs are excluded.
ER -