An integer partition of a number $n$ is a way of writing $n$ as a sum of positive integers.
Partitions that differ only in the order of their summands are considered the same. A partition of $n$ into distinct parts is a partition of $n$ in which every part occurs at most once.
The partitions of $5$ into distinct parts are:
$5$, $4+1$ and $3+2$.
Let $f(n)$ be the maximum product of the parts of any such partition of $n$ into distinct parts and let $m(n)$ be the number of elements of any such partition of $n$ with that product.
So $f(5)=6$ and $m(5)=2$.
For $n=10$ the partition with the largest product is $10=2+3+5$, which gives $f(10)=30$ and $m(10)=3$.
And their product, $f(10) \cdot m(10) = 30 \cdot 3 = 90$.
It can be verified that
$\sum f(n) \cdot m(n)$ for $1 \le n \le 100 = 1683550844462$.
Find $\sum f(n) \cdot m(n)$ for $1 \le n \le 10^{14}$.
Give your answer modulo $982451653$, the $50$ millionth prime.
solutions/s374.md.