A090086 Smallest pseudoprime to base n, not necessarily exceeding n (cf. A007535).

4, 341, 91, 15, 4, 35, 6, 9, 4, 9, 10, 65, 4, 15, 14, 15, 4, 25, 6, 21, 4, 21, 22, 25, 4, 9, 26, 9, 4, 49, 6, 25, 4, 15, 9, 35, 4, 39, 38, 39, 4, 205, 6, 9, 4, 9, 46, 49, 4, 21, 10, 51, 4, 55, 6, 15, 4, 57, 15, 341, 4, 9, 62, 9, 4, 65, 6, 25, 4, 69, 9, 85, 4, 15, 74, 15, 4, 77, 6, 9, 4, 9, 21, 85, 4, 15, 86, 87, 4, 91, 6

Offset: 1

Author

Labos Elemer, Nov 25 2003

Keywords

• nonn

Comments

If n-1 is composite, then a(n) < n. - Thomas Ordowski, Aug 08 2018

Conjecture: a(n) = A007535(n) for finitely many n. For n > 2; if a(n) > n, then n-1 is prime (find all these primes). - Thomas Ordowski, Aug 09 2018

It seems that if a(2^p) = p^2, then 2^p-1 is prime. - Thomas Ordowski, Aug 10 2018

a(n) is the smallest composite k such that n^(k-1) == (1-k)^n (mod k). - Thomas Ordowski, Mar 19 2025

Examples

			From _Robert G. Wilson v_, Feb 26 2015: (Start)
a(n) = 4 for n = 1 + 4*k, k >= 0.
a(n) = 6 for n = 7 + 12*k, k >= 0.
a(n) = 9 for n = 8 + 18*k, 10 + 18*k, 35 + 36*k, k >= 0.
(End)
a(n) = 10 for n = 51 + 60*k, 11 + 180*k, 131 + 180*k, k >= 0.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1024 terms from Eric Chen)
• Wikipedia, Pseudoprime
• Index entries for sequences related to pseudoprimes

Crossrefs

Cf. A007535, A250200, A090085, A090087, A000790, A239293, A293203.

Cf. A001567, A005935, A005936, A005937, A005938, A005939, A020136 - A020228.

Programs

Mathematica

f[n_] := Block[{k = 1}, While[ GCD[n, k] > 1 || PrimeQ[k] || PowerMod[n, k - 1, k] != 1, j = k++]; k]; Array[f, 91] (* Robert G. Wilson v, Feb 26 2015 *)

PARI

/* a(n) <= 2000 is sufficient up to n = 10000 */
a(n) = for(k=2,2000,if((n^(k-1))%k==1 && !isprime(k), return(k))) \\ Eric Chen, Feb 22 2015

PARI

a(n) = {forcomposite(k=2, , if (Mod(n,k)^(k-1) == 1, return (k)););} \\ Michel Marcus, Mar 02 2015

Formula

a(n) = LeastComposite{x; n^(x-1) mod x = 1}.