A233519 The number of times x^x == x (mod prime(n)) for x in 0 < x < prime(n).

1, 1, 1, 2, 1, 2, 3, 2, 2, 2, 3, 2, 5, 3, 2, 1, 1, 4, 3, 2, 5, 4, 1, 4, 5, 3, 4, 1, 2, 5, 5, 3, 5, 4, 2, 5, 7, 3, 2, 2, 1, 5, 3, 9, 3, 7, 6, 2, 1, 2, 6, 3, 8, 5, 5, 2, 1, 8, 3, 10, 2, 1, 8, 6, 9, 1, 9, 12, 1, 5, 5, 2, 4, 6, 6, 2, 1, 4, 5, 9, 4, 4, 3, 6, 4, 5, 6

Offset: 1

Author

T. D. Noe, Feb 19 2014

Keywords

• nonn

Comments

This is A065295 restricted to the primes. The plot is significantly different.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Pär Kurlberg, Florian Luca, and Igor Shparlinski On the fixed points of the map x -> x^x modulo a prime arX1v 1402.4464

Crossrefs

Cf. A065295, A233518, A233520.

Programs

Mathematica

f[p_] := Module[{x = Range[p - 1]}, Count[PowerMod[x, x, p] - x, 0]]; Table[f[n], {n, Prime[Range[100]]}]