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id	title	challengeType	forumTopicId	dashedName
5900f41c1000cf542c50ff2e	Problem 175: Fractions involving the number of different ways a number can be expressed as a sum of powers of 2	5	301810	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 \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}.

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.

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

\frac{f(n)}{f(n - 1)} = \frac{123456789}{987654321}

Give your answer as a string with comma separated integers, without any whitespaces.

--hints--

shortenedBinaryExpansionOfNumber() should return a string.

assert(typeof shortenedBinaryExpansionOfNumber() === 'string');

shortenedBinaryExpansionOfNumber() should return the string 1,13717420,8.

assert.strictEqual(shortenedBinaryExpansionOfNumber(), '1,13717420,8');

--seed--

--seed-contents--

function shortenedBinaryExpansionOfNumber() {

  return true;
}

shortenedBinaryExpansionOfNumber();

--solutions--

// solution required

1.8 KiB Raw Blame History