a remark on the factorization of factorials

The subject of this paper is to study distribution of the prime factors p and their exponents, which we denote by vp (n!), in standard factorization of n! into primes. we show that for each θ > 0 the primes p not exceeding nθ eventually assume almost all value of the sum ∑p⩽nθ vp(n!). also, we introduce the notion of θtruncated factorial, defined by n!θ =∏p⩽nθ  pvp (n!) and we show that the growth of log n!1/2 is almost half of growth of log n!1.