A360475 Smallest prime factor of (2^prime(n) + 1) / 3.

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 59, 715827883, 1777, 83, 2932031007403, 283, 107, 2833, 768614336404564651, 7327657, 56409643, 1753, 201487636602438195784363, 499, 179, 971, 845100400152152934331135470251, 415141630193, 643, 104124649, 227

Offset: 2

Author

Alain Rocchelli, Feb 08 2023

Keywords

• nonn

Comments

If (2^prime(n) + 1) / 3 is prime then a(n) is a Wagstaff prime (cf. A000979).

For n > 2, a(n) is congruent to 1 (mod 2*prime(n)).

Examples

			a(2)=3 since for prime(2)=3, (2^3+1)/3 = 3;
a(3)=11 since for prime(3)=5, (2^5+1)/3 = 11;
a(10)=59 since for prime(10)=29, (2^29+1)/3 = 59*3033169.
		

Crossrefs

Cf. A020639, A126614.

Programs

Maple

a:= n-> min(numtheory[factorset]((2^ithprime(n)+1)/3)):
seq(a(n), n=2..30);  # Alois P. Heinz, Feb 28 2023

Mathematica

a[n_] := FactorInteger[(2^Prime[n]+1)/3][[1, 1]];
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Jan 27 2025 *)

PARI

forprime(p=3, 100, An=(2^p+1)/3; if(isprime(An), print1(An,", "), forprime(div=3, 2^((p-1)/2), if(An%div==0, print1(div,", "); next(2)))))

Formula

a(n) = A020639(A126614(n)).

Extensions

a(26)-a(30) from Amiram Eldar, Feb 08 2023