Given a sequence of distinct positive integers $w_0 , w_1, w_2, \ldots$ and any positive integer $n$, we define the discriminator function $\mathcal{ D}_{ \bf w}(n)$ to be the smallest positive integer $m$ such that $w_0,\ldots, w_{ n-1}$ are pairwise incongruent modulo $m$. In this paper, we classify all binary recurrent sequences $\{ w_n\}_{ n\geq 0}$ consisting of different integer terms such that $\mathcal{ D}_{ \bf w}(2^e)=2^e$ for every $e\geq 1.$ For all of these sequences it is expected that one can actually give a fairly simple description of $\mathcal{ D}_{ \bf w}(n)$ for every $n\ge 1.$ For two infinite families of such sequences this has been done already in 2019 by Faye, Luca and Moree, respectively Ciolan and Moree.
Categories: math.NT