A065295 - OEIS

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

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OFFSET

1,4

COMMENTS

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

LINKS

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

Cf. A065296.

Sequence in context: A321088 A070556 A277687 * A296604 A261211 A233521

Adjacent sequences: A065292 A065293 A065294 * A065296 A065297 A065298

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved