On the prime power factorization of n!

For a fixed prime q, let eq(n) denote the order of q in the prime factorization of n!. For two fixed integers m⩾2 and r with 0⩽r⩽m−1, let A(x;m,q,r) denote the numbers of positive integers n⩽x for which . In this paper we shall prove a sharp asymptotic formula of A(x;m,q,r).