---

Statement

An integer $n\ge1$ is practical when every positive integer $m\le n$ is a sum of distinct divisors of $n$; $P(x)$ is the number of practical numbers not exceeding $x$ (p. 1). The paper recalls the criterion of Stewart and Sierpinski: an integer $n\ge2$ with prime factorization $n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}$, $p_1<\cdots<p_k$, is practical if and only if $p_j\le1+\sigma\bigl(\prod_{1\le i\le j-1}p_i^{\alpha_i}\bigr)$ for $1\le j\le k$, the empty product being $1$ (p. 1).

Theorem 1 (p. 1). There is a positive constant $c$ such that, for $x\ge3$,

$$P(x)=\frac{cx}{\log x}\Bigl\{1+O\Bigl(\frac{\log\log x}{\log x}\Bigr)\Bigr\}.$$

This settles Margenstern's conjecture that $P(x)$ is asymptotic to $cx/\log x$, and sharpens Saias's two-sided bound $c_1x/\log x\le P(x)\le c_2x/\log x$ for $x\ge2$, which the paper recalls on the same page. In particular the practical numbers have natural density zero.

Source. Andreas Weingartner, Practical numbers and the distribution of divisors, Q. J. Math. 66 (2015), no. 2, 743--758, read in arXiv:1405.2585v3 (3 March 2015), as identified on the source card; Theorem 1 on p. 1, in Section 1 (pp. 1--5). The labels are the preprint's; the published version was not compared.

Read depth. Claims checked: the statement and the criterion above were read clause by clause on the page image of p. 1, and the deduction from Theorem 2 on p. 2. The proof of Theorem 2 was read for its structure only.

Proof pointer

The paper deduces Theorem 1 from Theorem 2 (p. 2): with $\theta(n)=\sigma(n)+1$ the set $\mathcal B$ of Theorem 2 is the set of practical numbers, so $B(x)=P(x)$, and $\sigma(n)+1=O(n\log\log3n)$ puts $\theta$ in the range of Theorem 2 with $(a,b)=(0,1)$, whose error term is then $O((\log x)^{-1}\log\log x)$.

Bears on

• Problem 859: if $N$ is practical and $N\ge t$, then $t$ is a sum of distinct divisors of $N$, and so of every multiple of $N$. Theorem 1 counts the practical numbers themselves and says nothing about the density $d_t$ of the integers that represent a fixed $t$.
• Problem 673: the theorem gives $P(x)=o(x)$, so any statement proved only along practical numbers concerns a set of density zero and does not reach the problem's almost-all question. The paper says nothing about the sum $G(n)$ of consecutive divisor ratios.