A280338 Number of sizes of remainder sets for n, for any natural number c, given natural number b in (b^c) mod n.
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, 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, 4, 6, 8, 8, 12, 9, 7, 10, 8, 6, 8, 6, 9, 8, 4, 6, 5, 8
Offset: 1
Keywords
Examples
For a(1): b^c mod 1 = 0, so only 1 remainder set (0) is possible, and its size is 1. 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). 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). 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).
Links
- Jeptha Davenport, Table of n, a(n) for n = 1..500
Crossrefs
First differs from A062821 at index n=15.