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id	title	challengeType	forumTopicId	dashedName
5900f53b1000cf542c51004d	Problem 462: Permutation of 3-smooth numbers	5	302137	problem-462-permutation-of-3-smooth-numbers

--description--

A 3-smooth number is an integer which has no prime factor larger than 3. For an integer N, we define S(N) as the set of 3-smooth numbers less than or equal to N. For example, S(20) = \\{1, 2, 3, 4, 6, 8, 9, 12, 16, 18\\}.

We define F(N) as the number of permutations of S(N) in which each element comes after all of its proper divisors.

This is one of the possible permutations for N = 20.

1, 2, 4, 3, 9, 8, 16, 6, 18, 12.

This is not a valid permutation because 12 comes before its divisor 6.

1, 2, 4, 3, 9, 8, 12, 16, 6, 18.

We can verify that F(6) = 5, F(8) = 9, F(20) = 450 and F(1000) ≈ 8.8521816557e\\,21.

Find F({10}^{18}). Give as your answer as a string in its scientific notation rounded to ten digits after the decimal point. When giving your answer, use a lowercase e to separate mantissa and exponent. E.g. if the answer is 112\\,233\\,445\\,566\\,778\\,899 then the answer format would be 1.1223344557e17.

--hints--

permutationOf3SmoothNumbers() should return a string.

assert.strictEqual(typeof permutationOf3SmoothNumbers() === 'string');

permutationOf3SmoothNumbers() should return the string 5.5350769703e1512.

assert.strictEqual(permutationOf3SmoothNumbers(), '5.5350769703e1512');

--seed--

--seed-contents--

function permutationOf3SmoothNumbers() {

  return true;
}

permutationOf3SmoothNumbers();

--solutions--

// solution required

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