# Problem 374: Maximum Integer Partition Product
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)·m(10) = 30·3 = 90 It can be verified that ∑f(n)·m(n) for
1 ≤ n ≤ 100 = 1683550844462. Find ∑f(n)·m(n) for 1 ≤ n ≤ 1014. Give your
answer modulo 982451653, the 50 millionth prime.