A307864 - OEIS

%I #15 Jul 02 2019 13:39:47

%S 0,1,2,3,2,5,6,1,8,9,4,11,4,1,14,15,4,1,18,1,20,21,4,23,6,1,26,1,4,29,

%T 30,1,16,33,4,35,36,1,4,39,2,41,4,1,44,9,4,1,48,1,50,51,16,53,54,1,56,

%U 1,4,1,10,1,2,63,4,65,36,1,68,69,4,1,16,1,74,75,8,1,78,1,4,81,8,83,12,1,86

%N a(n) is the number of natural bases b < 2n+1 such that b^n == 1 (mod 2n+1).

%C For n > 0, a(n) = n if and only if 2n+1 is prime.

%C If a(n) < n, then 2n+1 is composite.

%C Theorem: a(n) > n if and only if 2n+1 is an absolute Euler pseudoprime.

%C Conjecture: if 2n+1 is an absolute Euler pseudoprime, then a(n) = phi(2n+1).

%t a[n_] := Length[Select[Range[2n], PowerMod[#, n, 2n+1] == 1 &]]; Array[a, 100] (* _Amiram Eldar_, May 02 2019 *)

%o (PARI) a(n) = sum(b=1, 2*n, Mod(b, 2*n+1)^n == 1); \\ _Michel Marcus_, May 02 2019

%Y Cf. A000010, A033181, A306530, A307865.

%K nonn

%O 0,3

%A _Thomas Ordowski_, May 02 2019

%E More terms from _Amiram Eldar_, May 02 2019