Take the 2-minute tour ×  I think this question is famous, but I
searched a lot in the internet with many keywords and
unfortunately I did not find any related problem. I will
appreciate any helpful answers and any guidance for similar
questions and problems.  Suppose the integers $n$ and $k$ are
given and the set $S$ shows all possible factorization of the
integer $n$ into $k$ factors. The factors of these factorization
not necessarily primes and can be composite. Also, at most one
factor can be the number $1$. For example, if $n=128$ and $k=3$,
we have:  $S=\lbrace 2\times4\times16, 4\times4\times8, 2\times
2\times32, 1\times2\times64,
1\times4\times32,1\times8\times16\rbrace$.  For fixed numbers
$n$ and $k$ (and also the set $S$), let $m\in S$ and
$m=l_1\times l_2 \ldots\times l_k$. We define
$S(m)=\sum_{i=1}^k{l_i}$.  What can we say about the minimum of
$S(m)$, where $m$ changes in the set $S$? Do we have any formula
for finding this minimum respect to the $n$ and $k$?  For the
above example, the minimum