A333339 a(n) is the smallest positive number k such that n divides 3^k - k.

1, 1, 3, 3, 7, 3, 2, 3, 9, 7, 4, 3, 16, 5, 27, 11, 5, 9, 29, 7, 27, 45, 39, 3, 73, 27, 27, 27, 22, 27, 132, 27, 36, 5, 27, 27, 65, 29, 27, 27, 27, 27, 10, 59, 27, 39, 12, 27, 47, 73, 42, 27, 68, 27, 36, 27, 30, 47, 154, 27, 192, 147, 27, 59, 16, 45, 119, 75, 39

Offset: 1

Author

Jinyuan Wang, Mar 16 2020

Keywords

• nonn

Comments

For any positive integer n, if k = a(n) + n*m*A007734(n) and m >= 0 then 3^k - k is divisible by n.

a(n) > log_3(n). - Robert Israel, Mar 19 2020

Links

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

Crossrefs

Cf. A007734, A072872, A333334.

Programs

Maple

f:= proc(n) local k;
  for k from 1 do if 3 &^k - k mod n = 0 then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Mar 19 2020

Mathematica

a[n_] := Module[{k = 1}, While[!Divisible[3^k - k, n], k++]; k]; Array[a, 100] (* Amiram Eldar, Mar 16 2020 *)

PARI

a(n) = for(k=1, oo, if(Mod(3, n)^k==k, return(k)));

Formula

a(3^m) = 3^m for m >= 0.

a(3^m-m) = m for m >= 1. - Robert Israel, Mar 19 2020