Školjka

%V0 Does there exist a sequence $F(1), F(2), F(3), \ldots$ of non-negative integers that simultaneously satisfies the following three conditions? (a) Each of the integers $0, 1, 2, \ldots$ occurs in the sequence. (b) Each positive integer occurs in the sequence infinitely often. (c) For any $n \geq 2,$ $$F(F(n^{163})) = F(F(n)) + F(F(361)).$$