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

1,1

Comments

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

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.

Example

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.

Mathematica

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

Prog

(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))
(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 fractions 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

Crossrefs

Cf. A066803, A098640, A349722.

Sequence in context: A140157 A285689 A139867 * A090434 A062603 A007353

Adjacent sequences: A260671 A260672 A260673 * A260675 A260676 A260677

Keyword

nonn

Author

Alex Jordan, Nov 14 2015

Status

approved