Problem 18 — Visual explainer (undergrad)
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Problem statement
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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\[h(m) < (\log\log m)^{O(1)}?\]Is it true that $h(n!)<n^{o(1)}$? Or perhaps even $h(n!)<(\log n)^{O(1)}$?

Picture
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Divisors / subset-sum picture:

divisors(n):  1, d1, d2, ..., n
choose some of them and add them up

σ(n) = sum of *all* divisors

Question: what to look at
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- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Context
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Area hint: multiplicative number theory.

Notation (if it appears above)
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- `o(g(n))` means "grows strictly smaller than g(n)" (ratio → 0 as n→∞).
- `O(g(n))` means "at most a constant times g(n)" for large n.

Benchmarks / known results (from comments)
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- Erd\H{o}s originally showed that $h(n!) <n$.
- Vose proved the existence of infinitely many practical $m$ such that
  $h(m)\ll (\log m)^{1/2}$.
- The reward of \$250 is offered in, apparently (although this is not entirely
  clear) for a proof or disproof of whether\[h(n!) <(\log n)^{O(1)}.\]See also
  [304] and [825].