A260674 - OEIS

%I #47 Dec 07 2021 11:08:15

%S 2,83,107,367,569,887,1327,1451,1621,1987,2027,3307,3547,3631,3691,

%T 4421,4547,4967,5669,5843,5927,6011,6911,6991,7207,7949,8167,8431,

%U 10771,10889,11287,11621,12007,12227,12487,12763,12983,15391,15767,16127,17107,17183,17231

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

%C Primes p such that A066803(p)>1. - _Tom Edgar_, Nov 15 2015

%H Chai Wah Wu, <a href="/A260674/b260674.txt">Table of n, a(n) for n = 1..1000</a>

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

%o (Sage)

%o # code will list all such primes no larger than the N-th prime.

%o N=1000

%o for k in range(N):

%o     if (gcd(2^Primes().unrank(k)+1,3^Primes().unrank(k)+1) != 1):

%o         print(Primes().unrank(k))

%o (PARI) list(lim)=forprime(p=2,lim,if(gcd(2^p+1,3^p+1)>1,print1(p, ", "))) \\ _Anders Hellström_, Nov 14 2015

%o (Python)

%o from sympy import prime

%o from fractions import gcd

%o 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 Cf. A066803, A098640, A349722.

%K nonn

%O 1,1

%A _Alex Jordan_, Nov 14 2015