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 n θ eventually assume almost all value of the sum Σ p⩽n vp(n!). Also, we introduce the notion of θ-truncated factorial, defined by n! θ = Π p⩽n θ pvp(n!), and we show that the growth of log n!12 is almost half of growth of log n!1.