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Updated Oct 7

Problem 400

Statement. For any $k\geq 2$ let $g_k(n)$ denote the maximum value of

(a_1+\cdots+a_k)-n

where $a_1,\ldots,a_k$ are integers such that $a_1!\cdots a_k! \mid n!$ . Can one show that

\sum_{n\leq x}g_k(n) \sim c_k x\log x

for some constant $c_k$ ? Is it true that there is a constant $c_k$ such that for almost all $n<x$ we have

g_k(n)=c_k\log x+o(\log x)?

Status. Open.

Source. erdosproblems.com/400, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #400, https://www.erdosproblems.com/400.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

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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.

Linked from (9)

Problem 401 Factorials and Binomial Coefficients factorials_binomials/li_2026_prime_power_rarefaction_density_one_lower Corollary 1.3 (p. 2): liminf and limsup of (1/(x log x)) sum_{n<=x} g_k(n) lie in [3(k-1)/log 12, (k-1)/log 2]Theorem 1.1 (p. 1): g_k(n) >= (3(k-1)/log 12 - eps) log n for all but o(x) integers n <= x Theorem 1.2 (p. 2): g_k(n) <= (k-1) log_2 n + log_2 log n + O_k(1)Theorem 1.4 (p. 2): normal order of base-p digit sums along Au + b with a growing S-unit A factorials_binomials/pomerance_2026_remarks_middle_binomial_coefficient factorials_binomials/sothanaphan_2026_resolution_erdos_problem_728_writeup_aristotle

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