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

1,1

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

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 *)

Prog

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

Crossrefs

Cf. A065190.

Sequence in context: A139712 A175856 A075365 * A075274 A331309 A178144

Adjacent sequences: A268463 A268464 A268465 * A268467 A268468 A268469

Keyword

nonn

Author

Thomas Ordowski, Feb 05 2016

Extensions

More terms from Michel Marcus, Feb 05 2016

Status

approved