On the Factorization of Integers

Abstract

The order of magnitude of the average of the exponents in the canonical factorization of an integer is discuissed. In particular, it is shown that this average has normal order one and a result which implies that the average order is one is also derived. Let n = plal * prar be the factorization of n as a product of powers of distinct primes. Our purpose is to consider the average of the exponents in this decomposition. Let co(n) = r; Q(n) = a, + * * * +ar; and a(n) =-(n)/c1(n). Since co(n) and Q(n) both have normal order log log n [1], it is obvious that a(n) has normal order one and it will be shown that a(n) has average order one. Similar results for the minimum and maximum exponents in the above factorization have been given recently by Niven [2]. LEMMA 1. Ensx 1/w(n) = O(x/log log x). PROOF. E 1/w(n) = E 1/w(n) + 1/@(n) ngx n5x; 2w (n) E [W(n) log log x]2 ngx; 2 (n) -!(log logX) 2 ? 1. n 2x;2 (n)<log log X