---
id: 5900f4b91000cf542c50ffcc
title: 'Problem 333: Special partitions'
challengeType: 5
forumTopicId: 301991
dashedName: problem-333-special-partitions
---

# --description--

All positive integers can be partitioned in such a way that each and every term of the partition can be expressed as 2ix3j, where i,j ≥ 0.

Let's consider only those such partitions where none of the terms can divide any of the other terms. For example, the partition of 17 = 2 + 6 + 9 = (21x30 + 21x31 + 20x32) would not be valid since 2 can divide 6. Neither would the partition 17 = 16 + 1 = (24x30 + 20x30) since 1 can divide 16. The only valid partition of 17 would be 8 + 9 = (23x30 + 20x32).

Many integers have more than one valid partition, the first being 11 having the following two partitions. 11 = 2 + 9 = (21x30 + 20x32) 11 = 8 + 3 = (23x30 + 20x31)

Let's define P(n) as the number of valid partitions of n. For example, P(11) = 2.

Let's consider only the prime integers q which would have a single valid partition such as P(17).

The sum of the primes q &lt;100 such that P(q)=1 equals 233.

Find the sum of the primes q &lt;1000000 such that P(q)=1.

# --hints--

`euler333()` should return 3053105.

```js
assert.strictEqual(euler333(), 3053105);
```

# --seed--

## --seed-contents--

```js
function euler333() {

  return true;
}

euler333();
```

# --solutions--

```js
// solution required
```
