1.8 KiB
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