A268466 Smallest m > 1 such that m^m == 1 (mod n).

2, 3, 2, 5, 4, 7, 6, 9, 8, 11, 5, 13, 3, 9, 4, 17, 4, 19, 9, 21, 8, 5, 22, 25, 24, 3, 26, 9, 7, 31, 6, 33, 10, 35, 6, 37, 9, 9, 8, 41, 10, 43, 6, 5, 8, 47, 46, 49, 18, 51, 4, 9, 13, 55, 12, 9, 20, 7, 29, 61, 15, 35, 8, 65, 8, 25, 22, 69, 22, 51, 5, 73, 18, 9, 26, 9, 12, 79, 24, 81

Offset: 1

Author

Thomas Ordowski, Feb 05 2016

Keywords

• nonn

Comments

For n > 1, a(n) <= n + (-1)^n = A065190(n).

Conjecture: a(n) = n + (-1)^n for infinitely many n.

If A002322(n) is coprime to n, then a(n) <= A002322(n).

From Robert Israel, Feb 05 2016: (Start)

For m > 1, a(n) = m iff n is a divisor of m^m - 1 that is not a divisor of k^k - 1 for 1 < k < m.

In particular, a(m^m - 1) = m.

Is there any m such that this is the only n for which a(n) = m? (End)

If n > m^m - 1, then a(n) > m. - Thomas Ordowski, Oct 20 2019

Links

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

Crossrefs

Cf. A065190.

Programs

Maple

f:= proc(n) local k;
for k from 2 do if igcd(k,n) = 1 and k &^ k mod n = 1 then return k fi od
end proc:
2,seq(f(n),n=2..100); # Robert Israel, Feb 05 2016

Mathematica

{2}~Join~Table[SelectFirst[Range[2, 1000], Mod[#^#, n] == 1 &], {n, 2, 80}] (* Michael De Vlieger, Feb 05 2016, corrected by Harvey P. Dale, Sep 10 2021 *)
smg1[n_]:=Module[{m=2},While[PowerMod[m,m,n]!=1,m++];m]; Join[{2},Array[ smg1,80,2]] (* Harvey P. Dale, Aug 13 2021 *)

PARI

a(n) = {my(m = 2); while (Mod(m,n)^m != Mod(1, n), m++); m;} \\ Michel Marcus, Feb 05 2016

Extensions

More terms from Michel Marcus, Feb 05 2016