Quadratic sequences with prime power discriminators

Abstract

The discriminator of an integer sequence $\text{s}=(s(i){)}_{i\ge 0}$, introduced by Arnold, Benkoski, and McCabe in 1985, is the function $D_{\text{s}}(n)$ that sends $n$ to the least integer $m$ such that the numbers $s(0),s(1),\dots ,s(n-1)$ 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 $\ge n$. We determine all such sequences for $p=2$, show that there are none for $p\ge 5$, and provide some partial results for $p=3$.

22 pages