It is easy to see that almost all numbers are not practical. Erd\H{o}s originally showed that $h(n!) <n$. Vose \cite{Vo85} proved the existence of infinitely many practical $m$ such that $h(m)\ll (\log m)^{1/2}$.

The sequence of practical numbers is A005153 in the OEIS.

The reward of \$250 is offered in \cite{Er81h}, 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].


References


[Er81h] Erd\H{o}s, P., Some problems and results on additive and multiplicative
number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.

[Vo85] Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24.