Problem 495

Writing n as the product of k distinct positive integers

Let W(n,k) be the number of ways in which n can be written as the product of k distinct positive integers.

For example, W(144,4) = 7. There are 7 ways in which 144 can be written as a product of 4 distinct positive integers:

Note that permutations of the integers themselves are not considered distinct.

Furthermore, W(100!,10) modulo 1 000 000 007 = 287549200.

Find W(10000!,30) modulo 1 000 000 007.

将n拆分成k个不重复正整数的乘积

用W(n,k)表示将n拆分成k个不重复正整数的乘积的方法数。

例如，W(144,4) = 7，即有7种方法将144拆分成4个不重复正整数的乘积：

注意正整数的不同排列不视为不同的拆分方法。

进一步的，我们有W(100!,10)除以1 000 000 007的余数为287549200。

求W(10000!,30)除以1 000 000 007的余数。