Let s(n) be the number of different remainders n mod k, where 1 ≤ k ≤ ⌊n/2⌋. This rather natural sequence is sequence A283190 in the OEIS, and although some basic facts are known, surprisingly, it has been barely studied. First, we prove the asymptotic formula s(n) = cn + O(n/(log n log log n)), where c is an explicit constant. Then we focus on the difference between the 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 upper bound on the difference is O(log log 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.