On a Smarandache multiplicative function and its parity

摘要

For any positive integer n, we define the Smarandache multiplicative function U(n)as follows: V(1) = 1. If n ＞1 and n = Pα11Pα22…Pαss denotes the factorization of n into prime powers, then U(n) = max{α1· P1, α2·P2,…,αs·ps}. The main purpose of this paper is using the elementary and analytic methods to study the parity of U(n), and give an interesting asymptotic formula for it.