Problem 1053

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Statement. Call a number $k$-perfect if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\log\log n)$?

Status. Open.

Source. erdosproblems.com/1053, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1053, https://www.erdosproblems.com/1053.

References.

• [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. Section B2, p. 78, records the multiperfect counts known at the end of 2002 and "Can $k$ be as large as we wish? Erdős conjectured that $k=o(\ln\ln n)$." Library home: guy_2004_unsolved_problems_number_theory.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• guy_2004_unsolved_problems_number_theory