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
Keywords
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
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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
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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.
Comments