---
id: 5900f41c1000cf542c50ff2e
title: >-
  Problem 175: Fractions involving the number of different ways a number can be
  expressed as a sum of powers of 2
challengeType: 5
forumTopicId: 301810
dashedName: >-
  problem-175-fractions-involving-the-number-of-different-ways-a-number-can-be-expressed-as-a-sum-of-powers-of-2
---

# --description--

Define f(0)=1 and f(n) to be the number of ways to write n as a sum of powers of 2 where no power occurs more than twice.

For example, f(10)=5 since there are five different ways to express 10:10 = 8+2 = 8+1+1 = 4+4+2 = 4+2+2+1+1 = 4+4+1+1

It can be shown that for every fraction p/q (p>0, q>0) there exists at least one integer n such that f(n)/f(n-1)=p/q. For instance, the smallest n for which f(n)/f(n-1)=13/17 is 241. The binary expansion of 241 is 11110001. Reading this binary number from the most significant bit to the least significant bit there are 4 one's, 3 zeroes and 1 one. We shall call the string 4,3,1 the Shortened Binary Expansion of 241. Find the Shortened Binary Expansion of the smallest n for which f(n)/f(n-1)=123456789/987654321. Give your answer as comma separated integers, without any whitespaces.

# --hints--

`euler175()` should return 1, 13717420, 8.

```js
assert.strictEqual(euler175(), 1, 13717420, 8);
```

# --seed--

## --seed-contents--

```js
function euler175() {

  return true;
}

euler175();
```

# --solutions--

```js
// solution required
```