---
id: 5900f4e51000cf542c50fff6
title: 'Problem 374: Maximum Integer Partition Product'
challengeType: 5
forumTopicId: 302036
dashedName: problem-374-maximum-integer-partition-product
---

# --description--

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.

# --hints--

`euler374()` should return 334420941.

```js
assert.strictEqual(euler374(), 334420941);
```

# --seed--

## --seed-contents--

```js
function euler374() {

  return true;
}

euler374();
```

# --solutions--

```js
// solution required
```