32fac23a2d
* fix: clean-up Project Euler 161-180 * fix: corrections from review Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
64 lines
1.8 KiB
Markdown
64 lines
1.8 KiB
Markdown
---
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id: 5900f41c1000cf542c50ff2e
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title: >-
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Problem 175: Fractions involving the number of different ways a number can be
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expressed as a sum of powers of 2
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challengeType: 5
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forumTopicId: 301810
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dashedName: >-
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problem-175-fractions-involving-the-number-of-different-ways-a-number-can-be-expressed-as-a-sum-of-powers-of-2
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---
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# --description--
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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.
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For example, $f(10) = 5$ since there are five different ways to express 10:
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$$10 = 8 + 2 = 8 + 1 + 1 = 4 + 4 + 2 = 4 + 2 + 2 + 1 + 1 = 4 + 4 + 1 + 1$$
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It can be shown that for every fraction $\frac{p}{q}\\; (p>0, q>0)$ there exists at least one integer $n$ such that $\frac{f(n)}{f(n - 1)} = \frac{p}{q}$.
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For instance, the smallest $n$ for which $\frac{f(n)}{f(n - 1)} = \frac{13}{17}$ is 241. The binary expansion of 241 is 11110001.
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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.
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Find the Shortened Binary Expansion of the smallest $n$ for which
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$$\frac{f(n)}{f(n - 1)} = \frac{123456789}{987654321}$$
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Give your answer as a string with comma separated integers, without any whitespaces.
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# --hints--
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`shortenedBinaryExpansionOfNumber()` should return a string.
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```js
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assert(typeof shortenedBinaryExpansionOfNumber() === 'string');
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```
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`shortenedBinaryExpansionOfNumber()` should return the string `1,13717420,8`.
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```js
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assert.strictEqual(shortenedBinaryExpansionOfNumber(), '1,13717420,8');
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```
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# --seed--
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## --seed-contents--
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```js
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function shortenedBinaryExpansionOfNumber() {
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return true;
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}
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shortenedBinaryExpansionOfNumber();
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```
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# --solutions--
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```js
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// solution required
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```
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