We call $m$ practical if every integer $1 \leq n < m$ is the sum of distinct divisors of $m$ . If $m$ is practical then let $h (m)$ be such that $h (m)$ many divisors always suffice.Are there infinitely many practical $m$ such that $h (m) < (lo g lo g m)^{O (1)} ?$ Is it true that $h (n!) < n^{o (1)}$ ? Or perhaps even $h (n!) < (lo g n)^{O (1)}$ ?

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

machinery: unit-fractions,g_kr-divisors,additive-combinatorics

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

$$ h(m):=\min\\{k:\ \forall,1\le N<m\ \exists\ \text{distinct divisors }d_1,\dots,d_t\mid m,\ t\le k,\ N=\sum_{i=1}^t d_i\\}. $$

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

3 LLM attack(s) recorded (gpt 5.2, gpt pro 5.2); unverified.

AMS 11 · test (literature)

theorem practicalH_one : practicalH 1 = 1

open