On the prime power factorization of n!

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In this paper we prove two results. The first theorem uses a paper of Kim K to show that for fixed primes p₁,...,pₖ, and for fixed integers m₁,...,mₖ, with pᵢ|mᵢ, the numbers (eₚ₁(n),...,eₚₖ(n)) are uniformly distributed modulo (m₁,...,mₖ), where eₚ(n) is the order of the prime p in the factorization of n!. That implies one of Sander's conjecture from S, for any set of odd primes. Berend B asks to find the fastest growing function f(x) so that for large x and any given finite sequence εᵢ∈ {0,1}, i≤ f(x), there exists n<x such that the congruences eₚᵢ(n)≡ εᵢ 2 hold for all i≤ f(x). Here, pᵢ is the ith prime number. In our second result, we are able to show that f(x) can be taken to be at least c₁ (x/(x)⁶)¹/⁹, with some absolute constant c₁, provided that only the first odd prime numbers are involved.