A307865 a(n) is the number of natural bases b < 2n+1 such that b^n == -1 (mod 2n+1).

0, 1, 2, 3, 0, 5, 6, 1, 8, 9, 0, 11, 0, 1, 14, 15, 0, 1, 18, 1, 20, 21, 0, 23, 0, 1, 26, 1, 0, 29, 30, 1, 0, 33, 0, 35, 36, 1, 0, 39, 0, 41, 4, 1, 44, 9, 0, 1, 48, 1, 50, 51, 0, 53, 54, 1, 56, 1, 0, 1, 0, 1, 2, 63, 0, 65, 0, 1, 68, 69, 0, 1, 0, 1, 74, 75, 0, 1, 78, 1, 0, 81, 0, 83, 0, 1, 86

Offset: 0

Author

Thomas Ordowski, May 02 2019

Keywords

• nonn

Comments

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

Note that a(n) < n if and only if 2n+1 is composite.

Conjecture: if 2n+1 is an absolute Euler pseudoprime, then a(n) = 0.

Crossrefs

Cf. A033181, A053760, A307864.

Programs

Mathematica

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

PARI

a(n) = sum(b=1, 2*n, Mod(b, 2*n+1)^n == -1); \\ Michel Marcus, May 02 2019

Extensions

More terms from Amiram Eldar, May 02 2019