A065295 Number of values of s, 0 < s <= n-1, such that s^s == s (mod n).

0, 1, 1, 2, 1, 4, 2, 4, 3, 4, 1, 7, 2, 5, 7, 6, 3, 8, 2, 9, 7, 5, 2, 13, 5, 8, 3, 11, 2, 14, 3, 6, 8, 8, 9, 13, 2, 7, 9, 17, 5, 18, 3, 11, 13, 5, 2, 19, 9, 12, 11, 13, 1, 8, 11, 18, 9, 7, 1, 27, 4, 7, 20, 10, 16, 18, 3, 13, 8, 21, 2, 23, 5, 6, 16, 14, 13, 23, 4, 27, 9, 11, 1, 31, 13, 10, 12, 20

Offset: 1

Author

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 28 2001

Keywords

• nonn

Comments

Note that a(1) = 0 by definition. - Harry J. Smith, Oct 15 2009

Examples

			For n=5 we have (1^1) mod 5 = 1, (2^2) mod 5 = 4, (3^3) mod 5 = 2, (4^4) mod 5 = 1. Only for s=1 does (s^s) mod 5=s, so a(5)=1.
		

Links

• Harry J. Smith and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Crossrefs

Cf. A065296.

Programs

Maple

a:= n-> add(`if`(s&^s-s mod n=0, 1, 0), s=1..n-1):
seq(a(n), n=1..88);  # Alois P. Heinz, Jun 09 2025

Mathematica

f[p_] := Module[{x = Range[p-1]}, Count[PowerMod[x, x, p] - x, 0]]; Table[f[n], {n, 100}] (* T. D. Noe, Feb 19 2014 *)

PARI

{ for (n=1, 1000, a=0; for (s=1, n - 1, if (s^s % n == s, a++)); if (n==1, a=0); write("b065295.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 15 2009

PARI

a(n) = sum(s=1, n-1, Mod(s, n)^s == s); \\ Michel Marcus, Jun 03 2025

Extensions

Definition revised by N. J. A. Sloane, Oct 15 2009.