Scientia Magna Vol. 6 (2010), No. 2, 123-131
Some identities involving function Ut(n) Caijuan Li Department of Mathematics, Northwest University, Xi’an, Shaanxi 710127, P.R.China Abstract In this paper, we use the elementary method to study the properties of pseudo Smarandache function Ut (n), and obtain some interesting identities involving function Ut (n), P and for any fixed integer n, offer a method of calculating of the infinite series ∞ i=1 Ut (n). Keywords Pseudo Smarandache function, some identities, elementary method.
§1. Introduction and result For any positive integer n and Ut (n) fixed t ≥ 1, we define function © ª Ut (n) = min 1t + 2t + 3t + · · · + nt + k, n | m, k ∈ N + , t ∈ N + , where n ∈ N + , m ∈ N + . Wang Yu studied the properties of pseudo Smarandache function Ut (n), and obtained calculation of the infinite series ∞ X i=1
Ut (1),
∞ X
Ut (2),
∞ X
Ut (3).
i=1
i=1
In this paper we use the elementary method to study the calculating of the infinite series ∞ X
Ut (n),
i=1
P∞ P∞ where n ≥ 4, obtain calculation of the infinite series i=1 Ut (4), Ut (5), and for any fixed P∞ i=1 integer n, we offer a method of calculating the infinite series i=1 Ut (n). Theorem 1. For any real positive integer s, we have ∞ X 1 1 1 1 1 1 1 1 1 1 1 s (n) = 1 + (2 − 2s − 3s + 5s )(1 − 5s ) + 7s (2 − 2s − 3s ) + (1 − 2s )(1 − 3s )(2 − 5s ). U i=1 4
Theorem 2. For any real positive integer s, we have ∞ X 2 1 1 1 s (n) = (2 − 2s + 4s − 6s )ζ(s). U i=1 5