Restricted Factorisations

 Published on Saturday, 8th September 2018, 10:00 pm; Solved by 157;
Difficulty rating: 90%

Problem 636

Consider writing a natural number as product of powers of natural numbers with given exponents, additionally requiring different base numbers for each power.

For example, $256$ can be written as a product of a square and a fourth power in three ways such that the base numbers are different.
That is, $256=1^2\times 4^4=4^2\times 2^4=16^2\times 1^4$

Though $4^2$ and $2^4$ are both equal, we are concerned only about the base numbers in this problem. Note that permutations are not considered distinct, for example $16^2\times 1^4$ and $1^4 \times 16^2$ are considered to be the same.

Similarly, $10!$ can be written as a product of one natural number, two squares and three cubes in two ways ($10!=42\times5^2\times4^2\times3^3\times2^3\times1^3=21\times5^2\times2^2\times4^3\times3^3\times1^3$) whereas $20!$ can be given the same representation in $41680$ ways.

Let $F(n)$ denote the number of ways in which $n$ can be written as a product of one natural number, two squares, three cubes and four fourth powers.

You are given that $F(25!)=4933$, $F(100!) \bmod 1\,000\,000\,007=693\,952\,493$,
and $F(1\,000!) \bmod 1\,000\,000\,007=6\,364\,496$.

Find $F(1\,000\,000!) \bmod 1\,000\,000\,007$.