Summary: A classical formula of Legendre gives the $p$-adic valuation for factorials as a finite sum of values of the floor function. This expression can be used to produce a formula for the $p$-adic valuation of $n$ as a finite sum of periodic functions. An analogous result is established for the $p$-adic valuation of the ASM numbers. This sequence counts the number of alternating sign matrices.

05A10, 11B75, 11Y55

alternating sign matrices, valuations, recurrences, digit count