A290542 a(n) is the least integer k in the interval [2, sqrt(n)] such that k^n == k (mod n), or 0 if no such integer exists.

0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 4, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 5, 3, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 0

Offset: 4

Author

Arkadiusz Wesolowski, Aug 05 2017

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 4..20000

Crossrefs

Cf. A010051, A290543.

Programs

Magma

lst:=[]; for n in [4..90] do r:=Floor(Sqrt(n)); for k in [2..r] do if Modexp(k, n, n) eq k then Append(~lst, k); break; end if; if k eq r then Append(~lst, 0); end if; end for; end for; lst;

Mathematica

Table[SelectFirst[Range[2, Sqrt@ n], PowerMod[#, n , n] == Mod[#, n] &] /. k_ /; MissingQ@ k -> 0, {n, 4, 90}] (* Michael De Vlieger, Aug 09 2017 *)

PARI

a(n) = for (k=2, sqrtint(n), if (Mod(k, n)^n == k, return(k));); return (0); \\ Michel Marcus, Aug 19 2017

Formula

a(A000040(n)) = 2 for n >= 3.

a(A001567(n)) = 2 for n >= 1.

a(A006935(n)) = 2 for n >= 2.

For n >= 3, a(x) = 2*A010051(x), where x = A000040(n).