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

1,3

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.

Links

Table of n, a(n) for n=1..90.

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)).

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 *)

Crossrefs

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

Sequence in context: A082119 A129708 A071518 * A065338 A316272 A294649

Adjacent sequences: A330339 A330340 A330341 * A330343 A330344 A330345

Keyword

nonn

Author

Thomas Ordowski, Dec 11 2019

Extensions

More terms from Amiram Eldar, Dec 11 2019

Status

approved