Let s (n) be the number of different remainders n k, where 1 k n/2. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has barely been studied. First, we prove that s (n) = c n + O (n/ (n n) ), where c is an explicit constant. Then we focus on differences between consecutive terms s (n) and s (n+1). It turns out that the value can always increase by at most one, but there exist arbitrarily large decreases. We show that the differences are bounded by O (n). Finally, we consider ''iterated remainder sets''. These are related to a problem arising from Pierce expansions, and we prove bounds for the size of these sets as well.
Baraskar et al. (Thu,) studied this question.