Scientia Magna Vol. 3 (2007), No. 2, 44-49
Some identities involving the near pseudo Smarandache function Yu Wang Department of Mathematics, Northwest University Xi’an, Shaanxi, P.R.China Received March 29, 2007 Abstract For any positive integer n and fixed integer t ≥ 1, we define function Ut (n) = min{k : 1t + 2t + · · · + nt + 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). Keywords Some identities, reciprocal, pseudo Smarandache function.
§1. Introduction and results In reference [1], A.W.Vyawahare defined the near pseudo Smarandache function K(n) as n(n + 1) K(n) = m = + k, where k is the small positive integer such that n divides m. Then he 2 studied the elementary properties of K(n), and obtained a series interesting results for K(n). n(n + 3) n(n + 2) For example, he proved that K(n) = , if n is odd, and K(n) = , if n is even; 2 2 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 ∞ X 1 1 function K(n), and proved that for any real number s > , the series is convergent, s (n) 2 K n=1 and ∞ X 1 2 5 = ln 2 + , K(n) 3 6 n=1 ∞ X n=1
1 K 2 (n)
=
11 2 22 + 2 ln 2 π − . 108 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, X n≤x
d(k) =
¶ ´ ³ 1 X µ n(n + 1) 3 = x log x + Ax + O x 2 log2 x , d K(n) − 2 4
n≤x