A216487 Smallest prime factor of n^(2n) - 1 having the form k*n+1.

3, 7, 5, 11, 7, 29, 17, 19, 11, 23, 13, 53, 29, 31, 17, 10949, 19, 108301, 41, 43, 23, 47, 73, 101, 53, 109, 29, 59, 31, 373, 257, 67, 103, 71, 37, 149, 191, 79, 41, 83, 43, 173, 89, 181, 47, 659, 97, 197, 101, 103, 53, 107, 109, 881, 113, 229, 59, 709, 61, 977

Offset: 2

Author

Michel Lagneau, Sep 11 2012

Keywords

• nonn

Comments

The corresponding values of k are in A216506.

Examples

			a(7) = 29 because 7^14 - 1 = 2 ^ 4 * 3 * 29 * 113 * 911 * 4733 and the smallest prime divisor of the form k*n+1 is 29 = 4*7+1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 2..670

Crossrefs

Cf. A006486, A007571, A187022, A187023, A216506.

Programs

Mathematica

Table[p=First/@FactorInteger[n^(2*n)-1]; Select[p, Mod[#1, n] == 1 &, 1][[1]], {n, 2, 41}]
a[n_] := Module[{m = n + 1}, While[!PrimeQ[m] || PowerMod[n, 2*n, m] != 1, m += n]; m]; Array[a, 100, 2] (* Amiram Eldar, May 17 2024 *)

PARI

a(n) = {my(m = n + 1); while(!isprime(m) || Mod(n, m)^(2*n) != 1, m += n); m;} \\ Amiram Eldar, May 17 2024

Formula

a(n) = Min{A187022(n), A187023(n)}.

Extensions

Data corrected by Amiram Eldar, May 17 2024