University of KashanMathematics Interdisciplinary Research2538-36396320210901A Remark on the Factorization of Factorials23524211189510.22052/mir.2021.240348.1254ENMehdiHassaniDepartment of Mathematics,
University of Zanjan,
University Blvd., 45371-38791
Zanjan, I. R. Iran0000-0003-0363-524XMahmoudMarieDepartment of Mathematics,
University of Zanjan,
University Blvd., 45371-38791
Zanjan, I. R. IranJournal Article20201105The subject of this paper is to study distribution of the prime factors p and their exponents, which we denote by v<sub>p</sub> (n!), in standard factorization of n! into primes. We show that for each θ > 0 the primes p not exceeding n<sup>θ</sup> eventually assume almost all value of the sum ∑<sub>p⩽n<sup>θ</sup> </sub>v<sub>p</sub>(n!). Also, we introduce the notion of θ-truncated factorial, defined by n!<sub>θ</sub> =∏<sub>p⩽n<sup>θ</sup> </sub> p<sup>v<sub>p</sub></sup> <sup>(n!) </sup>and we show that the growth of log n!<sub>1/2</sub> is almost half of growth of log n!1.https://mir.kashanu.ac.ir/article_111895_95cb4d730a26d08b2cc21d98d04876cb.pdf