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by Legendre's formula as follows:

:

Proof

Since is the product of the integers 1 through n, we obtain at least one factor of p in for each multiple of p in , of which there are . Each multiple of contributes an additional factor of p, each multiple of contributes yet another factor of p, etc. Adding up the number of these factors gives the infinite sum for .

Alternate form

One may also reformulate Legendre's formula in terms of the base-''p'' expansion

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