An Equidistribution Result for Differences Associated with Square Pyramidal Numbers

TL;DR

This paper derives an asymptotic formula for the average difference between square pyramidal numbers and their nearest squares, along with formulas for higher moments, advancing understanding of their distribution.

Contribution

It introduces new asymptotic formulas for the average and moments of differences between square pyramidal numbers and closest squares, a novel analysis in number theory.

Findings

Asymptotic formula for average difference $a_n$

Asymptotic formulas for the $k$-th moments

Enhanced understanding of distribution of differences

Abstract

We provide an asymptotic formula for the average value of the sequence A351830: $a_{n} = |P_{n} - y^{2}_{n}|$ for $1 \leq n \leq x$, where $P_{n}$ is the $n$-th square pyramidal number and $y^{2}_{n}$ is the closest square to $P_{n}$. Moreover, we supply asymptotic formulas for the $k$-th moment of the same sequence, for any fixed natural number $k$.