The board / number theory / #1053
Erdős Problem #1053
multiply perfect numbers
OPENdatabase updated 2025-09-28
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Call a number \(k\)-perfect if \(\sigma(n)=kn\), where \(\sigma(n)\) is the sum of the divisors of \(n\). Must \(k=o(\log\log n)\)? A question of Erdős, as reported in problem B2 of Guy's collection [Gu04]. The little-\(o\) relation is taken as \(n \to \infty\) within the positive integers for which \(\sigma(n) = kn\) for some \(k\); on this set, \(\sigma(n)/n\) is precisely that multiplier \(k\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1053.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA007539
Original pageerdosproblems.com/1053
Attempts
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Proved that #1053 is exactly equivalent to divergence of the missing-prime sum in (11), classified all squarefree cases, and exhaustively verified the 14 multiply-perfect numbers through \(10^8\); the
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