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Writing n as the product of k distinct positive integers

Problem 495 Published on Saturday, 27th December 2014, 10:00 pm; Solved by 187;
Difficulty rating: 100%

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:

  • 144 = 1×2×4×18
  • 144 = 1×2×8×9
  • 144 = 1×2×3×24
  • 144 = 1×2×6×12
  • 144 = 1×3×4×12
  • 144 = 1×3×6×8
  • 144 = 2×3×4×6

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.