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We say that $S\subset\mathbb Z$ is a set of $k$-recurrence if for everymeasure preserving transformation $T$ of a probability measure space $(X,\mu)$and every $A\subseteq X$ with $\mu(A)>0$, there is an $n\in S$ such that$\mu(A\cap T^{-n} A\cap T^{-2n}\cap \dots \cap T^{-kn}A)>0$. A set of$1$-recurrence is called a set of measurable recurrence.  Answering a question of Frantzikinakis, Lesigne, and Wierdl, we construct aset of $2$-recurrence $S$ with the property that $\{n^2:n\in S\}$ is not a setof measurable recurrence.

A set of 2-recurrence whose perfect squares do not form a set of measurable recurrence