On the number of infinite sequences with trivial initial segment complexity George Barmpalias, Tom Sterkenburg Abstract: The sequences which have trivial prefix-free initial segment complexity are known as K-trivial sets, and form a cumulative hierarchy of length ~. We show that the problem of finding the number of K-trivial sets in the various levels of the hierarchy is ~^0_3. This answers a question of Downey/Miller/Yu (see [DH10, Section 10.1.4]) which also appears in [Nie09, Problem 5.2.16]. We also show the same for the hierarchy of the low for K sequences, which are the ones that (when used as oracles) do not give shorter initial segment complexity compared to the computable oracles. In both cases the classification ~^0_3 is sharp.