Problem 955

Let

s(n)=\sigma(n)-n=\sum_{\substack{d\mid n\\ d<n}}d

be the sum of proper divisors function.

If $A\subset \mathbb{N}$ has density $0$ then $s^{-1}(A)$ must also have density $0$ .

Source: erdosproblems.com/955

Number theory

Wiki page

Status

Open

No claim settles this problem.

Open. The site labels the problem OPEN (page last edited 30 September 2025). Its commentary credits proofs for $A$ the primes, the integers with unusually many prime factors, the sums of two squares and the sets of size $x^{1/2+o(1)}$ , each an accepted partial claim; the general conjecture is open.

EGPS90: Erdős, P. and Granville, A. and Pomerance, C. and Spiro, C., On the normal behavior of the iterates of some arithmetic functions. Analytic number theory (Allerton Park, IL, 1989) (1990), 165-204.
Er73b: Erdős, P., Über die Zahlen der Form $\sigma (n)-n$ und $n-\phi(n)$ . Elem. Math. (1973), 83-86.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Part B has no passage stating the density-zero preimage assertion; the nearest is section B10 "Untouchable numbers", p. 100, which records that the untouchable numbers have positive lower density. Library home: Guy 2004.
PPT18: Pollack, Paul and Pomerance, Carl and Thompson, Lola, Divisor-sum fibers. Mathematika (2018), 330-342.
Po14b: Pollack, Paul, Some arithmetic properties of the sum of proper divisors and the sum of prime divisors. Illinois J. Math. (2014), 125-147.
Tr15: Troupe, Lee, On the number of prime factors of values of the sum-of-proper-divisors function. J. Number Theory (2015), 120-135.
Tr20: Troupe, Lee, Divisor sums representable as the sum of two squares. Proc. Amer. Math. Soc. (2020), 4189-4202.

Problem 955

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Proof claims

Discussion