--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 your answer as a string in 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