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# Minimum sum of fixed length factors of a number

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 is happen for $m=4\times4\times8$ and $S(m)=16$.

Is this a famous problem and are there any works related to this problem?

$Note:$ It can be seen that if the factors of $m\in S$ be as much possible as close to each other, then the sum of its factors converges to the minimum.