The least common multiple of consecutive terms in a quadratic progression

Authors

  • Shaofang Hong

Keywords:

quadratic progression, least common multiple, $p$-adic valuation, arithmetic function, the smallest period

Abstract

Let $k$ be any given positive integer. We define the arithmetic function $g_{k}$ for any positive integer $n$ by $g_{k}(n):=\frac{\prod_{i=0}^k ((n+i)^2+1)}{{\rm lcm}_{0\le i\le k}\{(n+i)^2+1\}}.$ We first show that the arithmetic function $g_{k}$ is periodic. Subsequently, we provide detailed local analysis to the periodic function $g_{k}$, and finally determine its smallest period. We also obtain an asymptotic formula for $\log {\rm lcm}_{0\le i\le k}\{(n+i)^2+1\}$. 10.1017/S0004972712000202

Published

2012-10-22

Issue

Vol. 86 No. 3 (2012)

Section

Articles
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