The board / number theory / #878
Erdős Problem #878
OPENdatabase updated 2025-08-31
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).
The function \(f(n)=\sum_{p\mid n}p^{\lfloor\log_p n\rfloor}\).
The function \(F(n)\) from Erdős Problem 878, with nontrivial terms \(a_i \ge 2\).
The maximum of an arithmetic function on \(\{0,\ldots,x\}\).
Is it true that, for almost all \(n\), \(f(n)=o(n\log\log n)\) and \(F(n)\gg n\log\log n\)? The answer is yes; a Lean proof is given in [Ki26].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 13 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/878.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA339378
Original pageerdosproblems.com/878
Attempts
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Live page status OPEN; 0 claimed proofs; currently working on this problem: KStar (live page accessed 2026-07-28).
SKIPPED