# Problem 462: Permutation of 3-smooth numbers
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(1018). 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.