Product of integers with known sum What is the biggest integer $N$ that can be expressed as product of some integers with sum $s$?
By AM-GM,
$$N\le \left(\frac{s}{n}\right)^n<e^{s/e}$$
Can we determine exactly $N$? If not, how can we improve the bound I got?

Write $S=s_1+s_2+\cdots+s_k$, with $\prod s_i$ maximal.  If any $s_i\ge 4$, replace it by $2+(s_i-2)$, which has product at least as large.  Hence we may assume each $s_i\in\{2,3\}$.  Note that $2+2+2=3+3$, while $2^3<3^2$, so there are at most $2$ $2$'s.  Thus, the answer is to take as many $3$'s as possible; if $S\equiv 0\pmod{3}$, then all $3$'s.  If $S\equiv 2\pmod{3}$, then one $2$ and the rest $3$'s.  If $S\equiv 1\pmod{3}$, then two $2$'s, and the rest $3$'s.
