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.
3-光滑数重排
3-光滑数是指所含质因数不大于3的整数。对于整数N，我们定义S(N)是所有小于等于N的3-光滑数所构成的集合，例如，S(20) = { 1, 2, 3, 4, 6, 8, 9, 12, 16, 18 }。
对集合S(N)的元素进行排列使得每个元素都排在其所有真因数之后，记这样的重排数为F(N)。
如下是N = 20时一个有效的重排。
1, 2, 4, 3, 9, 8, 16, 6, 18, 12.
如下不是一个有效的重排，因为12排在其真因数6之前。
1, 2, 4, 3, 9, 8, 12, 16, 6, 18.
可以验证F(6) = 5，F(8) = 9，F(20) = 450，以及F(1000) ≈ 8.8521816557e21。
求F(1018)，并用保留10位小数的科学计数法表示。
用小写字母e分割尾数和指数，例如，如果答案是112,233,445,566,778,899，则应写成1.1223344557e17。