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%I #31 Jan 02 2017 11:55:36
%S 1,1,2,2,3,2,4,3,4,3,4,3,6,4,3,3,5,4,6,5,4,4,4,4,6,6,6,6,6,3,8,6,4,5,
%T 6,6,9,6,6,6,8,4,8,7,7,4,4,5,8,6,5,9,6,6,6,8,6,6,4,5,12,8,6,7,6,4,8,9,
%U 4,6,8,8,12,9,7,10,8,6,8,6,9,8,4,6,5,8
%N Number of sizes of remainder sets for n, for any natural number c, given natural number b in (b^c) mod n.
%H Jeptha Davenport, <a href="/A280338/b280338.txt">Table of n, a(n) for n = 1..500</a>
%e For a(1): b^c mod 1 = 0, so only 1 remainder set (0) is possible, and its size is 1.
%e For a(2): for any b, b^c will be even if b is even, or odd if b is odd, so b^c mod 2 has only 1 remainder for a given b (either (0), size 1, or (1), also size 1).
%e For a(5): choosing c for an arbitrary b, for b = 2, 2^2 mod 5 = 4, 2^3 mod 5 = 3, 2^4 mod 5 = 1, 2^5 mod 5 = 2, 2^6 mod 5 = 4, etc. (4 remainders); for base 4, 4^1 mod 5 = 4, 4^2 mod 5 = 1, 4^3 mod 5 = 4, etc. (2 remainders); for base 21, 21^1 mod 5 = 1, 21^819 mod 5 = 1, etc. (1 remainder); these are the only numbers of remainders which occur for any c given b for b^c modulo 5, so the number of remainder set sizes for n = 5 is 3 (4, 2, or 1-size remainder sets).
%e For a(100): number of remainder set sizes possible for any c given b is 10 (1, 2, 3, 4, 5, 6, 10, 11, 20, or 21-size remainder sets).
%Y First differs from A062821 at index n=15.
%K nonn
%O 1,3
%A _Jeptha Davenport_, Dec 31 2016
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