Prime Factors of 1000 Factorial

Example of Factorial
The prime decomposition of $1000!$ is given as:

Proof
For each prime factor $p$ of $1000!$, let $a_p$ be the integer such that:
 * $p^{a_p} \mathrel \backslash 1000!$
 * $p^{a_p + 1} \nmid 1000!$

Taking the prime factors in turn:

Multiplicity of $13$
And so on: for the remaining factors:

The pattern is clear: the remaining indices are calculated in the same way.