## Permutation of 3-smooth numbers

### Problem 462

Published on Saturday, 8th March 2014, 07:00 pm; Solved by 154; Difficulty rating: 55%
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) ≈ 8.8521816557e21.

Find F(10^{18}). Give as your answer 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.