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

%I A260674 #51 Jun 22 2025 03:10:03
%S A260674 2,83,107,367,569,887,1327,1451,1621,1987,2027,3307,3547,3631,3691,
%T A260674 4421,4547,4967,5669,5843,5927,6011,6911,6991,7207,7949,8167,8431,
%U A260674 10771,10889,11287,11621,12007,12227,12487,12763,12983,15391,15767,16127,17107,17183,17231
%N A260674 Primes p for which the greatest common divisor of 2^p+1 and 3^p+1 is greater than 1.
%C A260674 Primes p such that A066803(p)>1. - _Tom Edgar_, Nov 15 2015
%H A260674 Chai Wah Wu, <a href="/A260674/b260674.txt">Table of n, a(n) for n = 1..1000</a>
%H A260674 Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_1064.htm">Puzzle 1064. GCD(2^p+1,3^p+1)</a>, The Prime Puzzles and Problems Connection.
%e A260674 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.
%t A260674 Select[Prime@ Range@ 2000, GCD[2^# + 1, 3^# + 1] > 1 &] (* _Michael De Vlieger_, Nov 16 2015 *)
%o A260674 (Sage)
%o A260674 # code will list all such primes no larger than the N-th prime.
%o A260674 N=1000
%o A260674 for k in range(N):
%o A260674     if (gcd(2^Primes().unrank(k)+1,3^Primes().unrank(k)+1) != 1):
%o A260674         print(Primes().unrank(k))
%o A260674 (PARI) list(lim)=forprime(p=2,lim,if(gcd(2^p+1,3^p+1)>1,print1(p, ", "))) \\ _Anders Hellström_, Nov 14 2015
%o A260674 (Python)
%o A260674 from sympy import prime
%o A260674 from math import gcd
%o A260674 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
%Y A260674 Cf. A066803, A098640, A349722.
%K A260674 nonn
%O A260674 1,1
%A A260674 _Alex Jordan_, Nov 14 2015