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Statement

A number $n$ is pseudoperfect (the paper follows Sierpiński, p. 1) when it is a sum of distinct proper divisors of $n$; $\sigma(n)/n$ is the abundance index of $n$.

Theorem 1 (p. 1, quoted): "For every $\varepsilon>0$ there exists an integer $L$ such that if $n$ is an integer with

$$\sigma(n)/n>2+\varepsilon$$

and it has no prime factors less than $L$, then $n$ is pseudoperfect."

$L$ depends on $\varepsilon$ only. The paper calls Theorem 1 its main theorem. Corollary 5 (p. 7) drops the condition on small prime factors and puts an absolute constant in place of $2+\varepsilon$ (Corollary 5).

Source. D. Larsen, Sufficiently abundant numbers are pseudoperfect, 9-page manuscript (GitHub Larsen-Daniel/Erdos-318, 318.pdf, commit 39139e2b of 1 February 2026); Theorem 1 on p. 1, proof on pp. 2--7.

Read depth. Claims checked: the statement was read clause by clause on the page image of p. 1. The proof was read for structure only (below) and is not verified here.

Proof pointer and sketch (pp. 2--7)

The paper first reduces to $\sigma(n)/n=O(1)$ (a multiple of a pseudoperfect number is pseudoperfect) and to squarefree $n$, and recasts the goal as writing $1$ as a sum of reciprocals of distinct divisors of $n$ greater than $1$. The prime factors of $n$ are grouped into dyadic ranges, keeping those dense enough, and the ranges are merged into blocks without large gaps. A greedy pass over the blocks stops at the first block whose reciprocal sum reaches the remaining distance to $1$ with a factor $1+\varepsilon/8$ to spare. Within that block, $D$ is the shortest initial part whose reciprocal sum $w(D)$ already has that spare factor. Lemma 2 (p. 4) then gives a subset $D_0$ of the divisors chosen so far (its proof removes at most one of them) such that $\alpha=w(D)/(1-w(D_0))$ lies in $[1+\varepsilon/100,\log^{1+\varepsilon}x_{a_{j_0}}]$. Theorem 4 (p. 5), with Hypothesis 3 checked for $\beta=1$ (p. 7), finds a subset of $D$ whose reciprocal sum is exactly $1-w(D_0)$, which ends the proof on p. 7.

Dependencies

Theorem 4 and Hypothesis 3 of the same paper (stated on p. 5, proof pp. 5--7, not checked here) and Lemma 2 (p. 4).

Standing

An unrefereed manuscript with a declared AI-assistance acknowledgment (proofreading, p. 9), read statically; no journal record was found on 2026-09-18. Consumers state the theorem with this qualification.

Bears on. #825: Theorem 1 is the paper's input to Corollary 5, which states an absolute $C$ such that every positive integer $n$ with $\sigma(n)\ge Cn$ is a sum of distinct proper divisors; Theorem 1 alone covers only integers with no prime factor below $L$.