The characteristic sequence of the integers that are the sum of two squares is not morphic

Abstract

Let $(s_2(n))_{n\in \mathbb{N}}$ be a $0,1$-sequence such that, for any natural number $n$, $s_2(n) = 1$ if and only if $n$ is a sum of two squares. In a recent article, Tahay proved that the sequence $(s_2(n))_{n\in \mathbb{N}}$ is not $k$-automatic for any integer $k$, and asked if this sequence can be morphic. In this note, we give a negative answer to this question.