Some identities involving the near pseudo Smarandache function

For any positive integer n and fixed integer t ≥ 1, we define function Ut(n) = min{k : 1 t + 2 t + · · · + n t + k = m, n | m, k ∈ N + , t ∈ N + }, where n ∈ N + , m ∈ N + , which is a new pseudo Smarandache function. The main purpose of this paper is using the elementary method to study the properties of Ut(n), and obtain some interesting identities involving function Ut(n). In reference [1], A.W.Vyawahare defined the near pseudo Smarandache function K(n) as K(n) = m = n(n + 1) 2 + k, where k is the small positive integer such that n divides m. Then he studied the elementary properties of K(n), and obtained a series interesting results for K(n). For example, he proved that K(n) = n(n + 3) 2 , if n is odd, and K(n) = n(n + 2) 2 , if n is even; The equation K(n) = n has no positive integer solution. In reference [2], Zhang Yongfeng studied the calculating problem of an infinite series involving the near pseudo Smarandache function K(n), and proved that for any real number s > 1 2 , the series ∞ n=1 1 K s (n) is convergent, and ∞ n=1 1 K(n) = 2 3 ln 2 + 5 6 , ∞ n=1 1 K 2 (n) = 11 108 π 2 − 22 + 2 ln 2 27. Yang hai and Fu Ruiqin [3] studied the mean value properties of the near pseudo Smarandache function K(n), and obtained two asymptotic formula by using the analytic method. They proved that for any real number x ≥ 1, n≤x d(k) = n≤x d K(n) − n(n + 1) 2 = 3 4 x log x + Ax + O x 1 2 log 2 x ,