Quadratic sequences with prime power discriminators

TL;DR

This paper characterizes quadratic sequences with a specific prime power discriminator, showing exact results for p=2, none for p≥5, and partial findings for p=3, advancing understanding of sequence discriminators.

Contribution

It identifies all quadratic sequences with prime power discriminators of the form p^{ceil(log_p n)} for p=2, none for p≥5, and offers partial results for p=3.

Findings

All such sequences are determined for p=2.

No such sequences exist for p≥5.

Partial results are provided for p=3.

Abstract

The discriminator of an integer sequence $\textbf{s} = (s(i))_{i \geq 0}$, introduced by Arnold, Benkoski, and McCabe in 1985, is the function $D_{\textbf{s}} (n)$ that sends $n$ to the least integer $m$ such that the numbers $s(0), s(1), \ldots, s(n - 1)$ are pairwise incongruent modulo $m$. In this note, we try to determine all quadratic sequences whose discriminator is given by $p^{\lceil \log_p n \rceil}$ for prime $p$, i.e., the smallest power of $p$ which is $\geq n$. We determine all such sequences for $p = 2$, show that there are none for $p \geq 5$, and provide some partial results for $p = 3$.