A191833 Least number k such that k^k == k+1 (mod m), or 0 if no such k exists, where m = A007310(n).

1, 7, 10, 14, 19, 11, 16, 3, 27, 43, 46, 178, 55, 36, 100, 64, 33, 79, 147, 43, 56, 258, 16, 86, 135, 52, 31, 27, 398, 335, 33, 187, 213, 151, 43, 680, 163, 61, 38, 243, 29, 327, 39, 213, 2068, 72, 37, 799, 198, 223, 141, 887, 92, 304, 132, 250, 808, 217, 327, 192, 271, 538, 398, 187, 79, 38, 31, 1713, 0, 413, 24, 1287, 976, 501, 48

Offset: 1

Author

Franklin T. Adams-Watters, Jun 17 2011

Keywords

• nonn

Comments

k^k == k+1 (mod m) does not have any solutions for m = 2 or 3, so only numbers in A007310 need be considered.

In general, if there is a solution, the first is less than m * phi(m), where phi is the Euler totient function A000010, since the values loop from that point (at least for units).

a(n) = 0 if and only if A007310(n) is in A191834. - Robert Israel, Sep 12 2017

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007310, A191834.

Programs

Maple

f:= proc(n) local m,k;
  m:= (6*n + (-1)^n - 3)/2;
  for k from 1 to ilcm(m,numtheory:-phi(m)) do
    if igcd(k,m) = 1 and k &^ k - k - 1 mod m = 0 then return k fi;
  od:
  0
end proc:
map(f, [$1..100]); # Robert Israel, Sep 12 2017

Mathematica

A007310[n_] := 2*n + 2*Floor[n/2] - 1; a[n_] := (For[m = A007310[n]; k = 1, k <= m^2, k++, If[PowerMod[k, k, m] == Mod[k+1, m], Return[k]]]; 0); Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Sep 13 2013 *)

PARI

a(n)=local(m);m=A007310(n);for(k=1,m^2,if(Mod(k,m)^k==k+1,return(k)));0