A330342 a(n) is the smallest k such that b^(n-1) == b^k (mod n) for all integers b.

0, 1, 2, 3, 4, 1, 6, 3, 2, 1, 10, 3, 12, 1, 2, 7, 16, 5, 18, 3, 2, 1, 22, 3, 4, 1, 8, 3, 28, 1, 30, 7, 2, 1, 10, 5, 36, 1, 2, 3, 40, 5, 42, 3, 8, 1, 46, 7, 6, 9, 2, 3, 52, 17, 14, 7, 2, 1, 58, 3, 60, 1, 2, 15, 4, 5, 66, 3, 2, 9, 70, 5, 72, 1, 14, 3, 16, 5, 78, 7, 26, 1, 82, 5, 4, 1, 2, 7, 88, 5

Offset: 1

Author

Thomas Ordowski, Dec 11 2019

Keywords

• nonn

Comments

Note that (n-1) == a(n) (mod lambda(n)), where lambda(n) = A002322(n).

For n > 1, a(n) = lambda(n) if and only if n is a prime or a Carmichael number. For n <> 1 and 4, a(n) = n-1 if and only if n is a prime.

For n > 2, a(n) = 1 if and only if n is a squarefree 2-Knodel number.

For n > 3, a(n) = 2 if and only if n is a 3-Knodel number.

Crossrefs

Cf. A000040, A002997, A002322, A033553, A050990, A051903, A276976.

Programs

Mathematica

a[n_] := Module[{k = 0}, While[!AllTrue[Range[n], PowerMod[#, n - 1, n] == PowerMod[#, k, n] &], k++]; k]; Array[a, 100] (* Amiram Eldar, Dec 11 2019 *)

Formula

a(n) = A(n) if A(n) >= A051903(n) or a(n) = A002322(n) + A(n) otherwise, where A(n) = ((n-1) mod A002322(n)).

Extensions

More terms from Amiram Eldar, Dec 11 2019