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
Keywords
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.
Programs
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Mathematica
Select[Prime@ Range@ 2000, GCD[2^# + 1, 3^# + 1] > 1 &] (* Michael De Vlieger, Nov 16 2015 *)
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PARI
list(lim)=forprime(p=2,lim,if(gcd(2^p+1,3^p+1)>1,print1(p, ", "))) \\ Anders Hellström, Nov 14 2015
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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
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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))
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