Let $n$ be any positive integer.
Let $p_1,p_2,...,p_m$ be any positive integers such that no more than one of the $p_i$s is $1$ and $\prod_{i=1}^mp_i=n$.
Finally, let $s_1,s_2,...,s_m$ be any nonnegative integers such that $\sum_{i=1}^ms_i=2n$.
Do there always exist nonnegative* integers $k_1,k_2,...,k_m$ such that: $$\sum_{i=1}^m(k_ip_i)=n$$
And, for each $i$: $$k_ip_i\leq s_i$$
?
Initially, I tried induction on $m$ (the number of factors, $p_i$). I assumed the hypothesis held for all cases when $m$ was less than $t$ for a fixed $n$ and tried proving it for the case of $m=t$ (for the same $n$) but got nowhere.
I also tried induction on the number of (not necessarily distinct) prime factors of $n$ but, again, only got past the base case.
(I couldn't really get past viewing this through an induction lens)
It seems as if it should be true given that the $s_i$'s need to sum to $2n$ whereas the $k_ip_i$'s need to sum to only $n$ (and the size of the $p_i$ terms are limited by the constraint that their product equal $n$) but I can't find a way to prove it (or a counterexample).
Are there any hints anyone can provide for a proof (or for generating a good counterexample)?
*EDIT: Originally, this question asked only about the existence of integer $k_i$s - the nonnegativity of these values was not considered, but it was expected that nonnegativity would emerge from a proof of the original question. A solution to the original question was found, but the approach taken did not seem to provide any leads for showing nonnegativity.