The density of sums of distinct divisors

Abstract

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 record the explicit bounds \[ \nu_t \le d_t \ll \frac{(\log\log t)^{\delta-3/2}}{(\log t)^{\delta}}, \qquad \nu_t = \frac{K}{\log t}\Bigl(1 + O\Bigl(\frac{\log\log t}{\log t}\Bigr)\Bigr), \] where $\delta = 0.086071\ldots$ is the Erd\H{o}s--Ford--Tenenbaum constant, $K=c\,e^{-\gamma}$, and $c=1.33607\ldots$ is the practical-number constant. Consequently, if Erd\H{o}s's asymptotic holds, then $\delta<c_4\le 1$. A two-prime construction, using a half-scale sumset to obtain full residue coverage, then yields the pointwise excess \[ \liminf_{t\to\infty}(\log t)\,(d_t-\nu_t) \ge KI, \] where $I=\int_0^2 G(w)\,\mathrm{d}w=0.05887\ldots$ is an explicit elementary integral. In particular $d_t\ge 0.79/\log t$ for every sufficiently large $t$, and $d_t$ is not asymptotic to $K/\log t$.