A301916 Primes which divide numbers of the form 3^k + 1.

2, 5, 7, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 73, 79, 89, 97, 101, 103, 113, 127, 137, 139, 149, 151, 157, 163, 173, 193, 197, 199, 211, 223, 233, 241, 257, 269, 271, 281, 283, 293, 307, 317, 331, 337, 349, 353, 367, 373, 379, 389, 397, 401, 409, 439

Offset: 1

Author

Luke W. Richards, Mar 28 2018

Keywords

• nonn

Comments

This sequence can be used to factor P-1 values for prime candidates of the form 3^k+2, to aid with primality testing.

a(1) = 2 divides every number of the form 3^k+1. It is the only term with this property.

For k > 2, A000040(k) is a member if and only if A062117(k) is even. - Robert Israel, May 23 2018

Examples

			Every value of 3^k+1 is an even number, so 2 is in the sequence.
No values of 3^k+1 is ever a multiple of 3 for any integer k, so 3 is not in the sequence.
3^2+1 = 10, which is a multiple of 5, so 5 is in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A034472, A062117, A301917, A320481.

Programs

Maple

f:= p -> numtheory:-order(3,p)::even:
f(2):= true:
select(isprime and f, [2,seq(p,p=5..1000,2)]); # Robert Israel, May 23 2018

Mathematica

Join[{2}, Select[Range[5, 1000, 2], PrimeQ[#] && EvenQ@ MultiplicativeOrder[3, #]&]] (* Jean-François Alcover, Feb 02 2023 *)

PARI

isok(p)=if (p != 3, m = Mod(3, p); nb = znorder(m); for (k=1, nb, if (m^k == Mod(-1, p), return(1)););); return(0); \\ Michel Marcus, May 18 2018

PARI

list(lim)=my(v=List([2]),t); forfactored(n=4,lim\1+1, if(n[2][,2]==[1]~, my(p=n[1],m=Mod(3,p)); for(k=2,znorder(m,t), m*=3; if(m==-1, listput(v,p); break))); t=n); Vec(v) \\ Charles R Greathouse IV, May 23 2018

PARI

isok(p)=isprime(p)&&if(p<4,p==2,znorder(Mod(3,p))%2==0) \\ Jeppe Stig Nielsen, Jun 27 2020

PARI

isok(p)=!isprime(p)&&return(0); p<4&&return(p==2); s=valuation(p-1,2); Mod(3,p)^((p-1)>>s)!=1 \\ Jeppe Stig Nielsen, Jun 27 2020