The board / number theory / #1054
Erdős Problem #1054
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).
Let \(f(n)\) be the minimal integer \(m\) such that \(n\) is the sum of the \(k\) smallest divisors of \(m\) for some \(k\geq 1\).
Let \(f(n)\) be the minimal integer \(m\) such that \(n\) is the sum of the \(k\) smallest divisors of \(m\) for some \(k\geq 1\). Is it true that \(f(n)=o(n)\)? The strong claim that \(f(n)=o(n)\) was disproved by Tao in the comments to [468](https://www.erdosproblems.com/468), in which he proves that the upper density of \(\{ n : f(n)\leq \delta n\}\) is \(\ll \delta^2\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1054.Prizenone
Topicsnumber theory, divisors
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA167485
Original pageerdosproblems.com/1054
Attempts
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Live page status is OPEN, but it lists jif and Vjeko_Kovac as currently working on problem #1054 (accessed 2026-07-28), so the mandatory no-collision rule requires stopping.
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