On a problem of the pseudo Smarandache function

For any positive integer n, the famous pseudo Smarandache function Z(n) is defined as the smallest positive integer m such that n|(m(m+1))/2.The main purpose of this paper is using the elementary method to find all positive integer n such that Z(n) is a primitive root of n. This problem proposed by Kenichiro Kashihara in his book [3], we solved it completely, and proved that Z(n) is a primitive root of n if and only if n = 2, 3, 4.