Factorial Fine Factorisations

A factorisation of N in factors is fine if each factor contains only 1 prime number:

6! = 720 possesses 10 fine factorisations:

• 5 9 16
• 3 3 5 16
• 2 5 8 9
• 2 3 3 5 8
• 4 4 5 9
• 3 3 4 4 5
• 2 2 4 5 9
• 2 2 3 3 4 5
• 2 2 2 2 5 9
• 2 2 2 2 3 3 5

How many fine factorisations are there for 10000!

Give the digital sum of this large number.

[My timing : 1 sec]