45 lines
1.4 KiB
Markdown
45 lines
1.4 KiB
Markdown
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---
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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 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 10:10 = 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 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.
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# --hints--
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`euler175()` should return 1, 13717420, 8.
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```js
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assert.strictEqual(euler175(), 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 euler175() {
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return true;
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}
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euler175();
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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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