Journal of Numbers http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. On the Distribution of -Tuples of -Free Numbers Sun, 13 Apr 2014 00:00:00 +0000 http://www.hindawi.com/journals/jn/2014/537606/ For k, being a fixed integer ≥2, a positive integer n is called k-free number if n is not divisible by the kth power of any integer >1. In this paper, we studied the distribution of r-tuples of k-free numbers and derived an asymptotic formula. Ting Zhang and Huaning Liu Copyright © 2014 Ting Zhang and Huaning Liu. All rights reserved. Sums of Products Involving Power Sums of Integers Thu, 27 Feb 2014 16:34:23 +0000 http://www.hindawi.com/journals/jn/2014/158351/ A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums which are defined via the Möbius function μ and the usual power sum of a real or complex variable . The power sum is expressible in terms of the well-known Bernoulli polynomials by . Jitender Singh Copyright © 2014 Jitender Singh. All rights reserved.