Question 23
S1 (UKR). Does there exist a sequence $F(1),F(2),F(3),\ldots$ of nonnegative 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 $n\ge2$,
$$F(F(n^{163}))=F(F(n))+F(F(361)).$$
(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 $n\ge2$,
$$F(F(n^{163}))=F(F(n))+F(F(361)).$$