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Let k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetical function g k,a,b for any positive integer n by g k,a,b (n) := (b+na)(b+(n+1)a)···(b+(n+k)a) lcm(b+na,b+(n+1)a,··· ,b+(n+k)a). Letting a = 1 and b = 0, then g k,a,b becomes the arithmetical function introduced previously by Farhi. Farhi proved g k,1,0 is periodical and k! is a period. divides(More)
For relatively prime positive integers u 0 and r, we consider the arithmetic progression {u k := u 0 + kr} n k=0. Define Ln := lcm{u 0 , u 1 ,. .. , un} and let a ≥ 2 be any integer. In this paper, we show that, for integers α, r ≥ a and n ≥ 2αr, we have Ln ≥ u 0 r α+a−2 (r + 1) n. In particular, letting a = 2 yields an improvement to the best previous(More)
Let p be a prime. We obtain good bounds for the p-adic sizes of the coefficients of the divided universal Bernoulli numberˆB n n when n is divisible by p − 1. As an application, we give a simple proof of Clarke's 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo p for the divided universal Bernoulli numbers for the(More)
In this paper, we investigate the 2-adic valuation of the Stirling numbers S(n, k) of the second kind. We show that v 2 (S(2 n + 1, k + 1)) = s 2 (n) − 1 for any positive integer n, where s 2 (n) is the sum of binary digits of n. This confirms a conjecture of Amdeberhan, Manna and Moll. We show also that v 2 (S(4i, 5)) = v 2 (S(4i + 3, 5)) if and only if i(More)
Nonlinearity of rotation symmetric Boolean functions is an important topic on cryptography algorithm. Let e ≥ 1 be any given integer. In this paper, we investigate the following question: Is the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial x 0 xex 2e x 3e equal to its weight? We introduce some new simple(More)