A366930 a(n) is the smallest odd composite k such that n^((k+1)/2) == n (mod k).

9, 9, 341, 121, 341, 65, 15, 21, 9, 9, 9, 33, 33, 21, 21, 15, 15, 9, 9, 9, 21, 15, 21, 33, 25, 15, 9, 9, 9, 21, 15, 15, 25, 33, 21, 9, 9, 9, 57, 39, 15, 21, 21, 21, 9, 9, 9, 65, 21, 21, 21, 15, 39, 9, 9, 9, 21, 21, 57, 145, 15, 15, 9, 9, 9, 33, 15, 33, 25, 21

Offset: 0

Author

Thomas Ordowski, Nov 01 2023

Keywords

• nonn

Comments

If this sequence is bounded, then it is periodic with period P = LCM(A), where A is the set of all (pairwise distinct) terms.

Note that n^((1729+1)/2) == n (mod 1729) for every n >= 0, where 1729 is the smallest absolute Euler pseudoprime (A033181).

Thus a(n) <= 1729. So, as said, this sequence is periodic.

What is its period?

The period P of this sequence may be longer than the period of Euler primary pretenders (A309316), namely P > 41#*571#/4 (248 digits).

Crossrefs

Cf. A033181, A309316, A366973, A366982.

Programs

Mathematica

a[n_] := Module[{k = 9}, While[PrimeQ[k] || PowerMod[n, (k + 1)/2, k] != Mod[n, k], k += 2]; k]; Array[a, 100, 0] (* Amiram Eldar, Nov 01 2023 *)

PARI

a(n) = my(k=3); while (isprime(k) || Mod(n, k)^((k+1)/2) != n, k+=2); k; \\ Michel Marcus, Nov 01 2023

Formula

a(n) >= A309316(n).