THE LEAST COMMON MULTIPLE OF CONSECUTIVE TERMS IN A QUADRATIC PROGRESSION

Let k be any given positive integer. We define the arithmetic function gk 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 gk is periodic. Subsequently, we provide a detailed local analysis of the periodic function gk, and determine its smallest period. We also obtain an asymptotic formula for log lcm0≤i≤k {(n+i)2+1}.