Let t(n) be the largest possible a_1 among factorizations n!=a_1… a_n with a_1≤…≤ a_n. Determine the second-order behavior of t(n): in particular, beyond t(n)/n→1/e, is t(n)≤ n/e-cn/log n infinitely often for some c>0?

Erdős Problem #391 — The smallest factor in an ordered factorization of a factorial

About 46 years old · traced to 1980

Let $t(n)$ be the largest possible $a_1$ among factorizations $n!=a_1\cdots a_n$ with $a_1\leq\cdots\leq a_n$ . Determine the second-order behavior of $t(n)$ : in particular, beyond $t(n)/n\to1/e$ , is $t(n)\leq n/e-cn/\log n$ infinitely often for some $c>0$ ?

References

Primary source

P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique 28 (1980).

Additional references

P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique 28 (1980).

factorials number-theory

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