A260674 Primes p for which the greatest common divisor of 2^p+1 and 3^p+1 is greater than 1.

2, 83, 107, 367, 569, 887, 1327, 1451, 1621, 1987, 2027, 3307, 3547, 3631, 3691, 4421, 4547, 4967, 5669, 5843, 5927, 6011, 6911, 6991, 7207, 7949, 8167, 8431, 10771, 10889, 11287, 11621, 12007, 12227, 12487, 12763, 12983, 15391, 15767, 16127, 17107, 17183, 17231

Offset: 1

Author

Alex Jordan, Nov 14 2015

Keywords

• nonn

Comments

Primes p such that A066803(p)>1. - Tom Edgar, Nov 15 2015

Examples

			Since GCD(2^83 + 1, 3^83 + 1) = 499, the prime 83 is in the sequence. It is only the second such prime, so a(2) = 83.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1000
• Carlos Rivera, Puzzle 1064. GCD(2^p+1,3^p+1), The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A066803, A098640, A349722.

Programs

Mathematica

Select[Prime@ Range@ 2000, GCD[2^# + 1, 3^# + 1] > 1 &] (* Michael De Vlieger, Nov 16 2015 *)

PARI

list(lim)=forprime(p=2,lim,if(gcd(2^p+1,3^p+1)>1,print1(p, ", "))) \\ Anders Hellström, Nov 14 2015

Python

from sympy import prime
from math import gcd
A260674_list = [p for p in (prime(n) for n in range(1,10**3)) if gcd(2**p+1,3**p+1) > 1] # Chai Wah Wu, Nov 23 2015

Sage

# code will list all such primes no larger than the N-th prime.
N=1000
for k in range(N):
    if (gcd(2^Primes().unrank(k)+1,3^Primes().unrank(k)+1) != 1):
        print(Primes().unrank(k))