## Maximum Integer Partition Product

### Problem 374

Published on Saturday, 3rd March 2012, 07:00 pm; Solved by 348

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.