We prove that for any nonnegative integers n and r the binomial sum n ∑ k=−n ( 2n n− k ) k is divisible by 22n−min{α(n),α(r)}, where α(n) denotes the number of 1s in the binary expansion of n. This confirms a recent conjecture of Guo and Zeng [J. Number Theory, 130(2010), 172–186]. In 1976 Shapiro [3] introduced the Catalan triangle ( k n ( 2n n−k ) )n>k>1 and determined the sum of entries in t...

Abstract. Let Rk(n) be the number of representations of an integer n as the sum of a prime and a k-th power for k ≥ 2. Furthermore, set Ek(X) = |{n ≤ X, n ∈ Ik, n not a sum of a prime and a k-th power}|. In the present paper we use sieve techniques to obtain a strong upper bound on Rk(n) for n ≤ X with no exceptions, and we improve upon the results of A. Zaccagnini to prove Ek(X) ≪k X 1−181 log...

Traditionally, clustering problems are investigated under the assumption that all objects must be clustered. A shortcoming of this formulation is that a few distant objects , called outliers, may exert a disproportionately strong influence over the solution. In this work we investigate the k-min-sum clustering problem while addressing outliers in a meaningful way. Given a complete graph G = (V,...

In 1631, Johannes Faulhaber published the result that sums of form $$\sum\limits_{i = 1}^n {{i^k}} {1^k} + {2^k} \,...\, {n^k}$$

Given a sequence of n real numbers and an integer k, 1 k 1 2n(n − 1), the k maximum-sum segments problem is to locate the k segments whose sums are the k largest among all possible segment sums. Recently, Bengtsson and Chen gave an O(min{k + n log2 n, n √ k})-time algorithm for this problem. Bae and Takaoka later proposed a more efficient algorithm for small k. In this paper, we propose an O(n ...