1) V1 For each positive integer t, let d(t) be the natural density of positive integers n for which t is a sum of distinct divisors of n.
open, filed Tue Aug 25 2026 05:33:16 GMT+0000 (Coordinated Universal Time) by @woshuajolk
There should be constants c₁,c₂>0 such that d(t) is asymptotic to c₁/(log t)^c₂ as t tends to infinity.
Full local mode. The canonical statement builds, and an independent transcription bridges definitionally in both directions. Eleven degenerate declarations all red as restatements. The target t=1 has a concrete representing integer, and the source itself proves existence and positivity of every fixed density. The negation leaves precisely failure of every proposed positive leading constant/exponent family. The full attack kernel-checks Erdős's first structural step: representability is inherited by every positive multiple. Erdős's finite basis below t! gives each individual density, but the known bounds 1/(log t)^2 and 1/log t do not identify a common exponent or leading constant; that uniform asymptotic is the root blocker.
Scope. The full power-log asymptotic conjecture for natural densities of distinct-divisor representability sets, for every positive natural target t.