A270249 Greater of a pair of twin primes (r,s=r+2) where s is of the form p^2 + pq + q^2 and p and q are also twin primes.

109, 433, 2056753, 3121201, 3577393, 26462701, 37340353, 43823053, 128786113, 202705201, 304093873, 888345793, 1005988033, 1399680001, 1537437133, 2282300173, 2310187501, 2444964913, 2929312513, 3564542701, 5831255233, 7950571201, 8512439473, 9346947373, 9648752833, 12627464653, 15624660673

Offset: 1

Author

Altug Alkan, Mar 14 2016

Keywords

• nonn

Comments

Subsequence of A243761.

How is the distribution of terms of this sequence? With this form p^2 + pq + q^2, do twin primes generate bigger twin primes infinitely many times?

Examples

			109 is a term because 109 and 107 are twin primes and 109 = 5^2 + 5*7 + 7^2, 5 and 7 are also twin primes.
433 is a term because 433 and 431 are twin primes and 433 = 11^2 + 11*13 + 13^2, 11 and 13 are also twin primes.
		

Crossrefs

Cf. A001359, A243761.

Programs

PARI

t(n, p=3) = {while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}
for(n=1, 1e3, if(ispseudoprime(P=(3*t(n)^2 + 6*t(n) + 4)) && ispseudoprime(P-2), print1(P, ", ")));

Python

from itertools import islice
from sympy import isprime, nextprime
def A270249_gen(): # generator of terms
    p, q = 2, 3
    while True:
        if q-p == 2 and isprime(s:=3*p*q+4) and isprime(s-2):
            yield s
        p, q = q, nextprime(q)
A270249_list = list(islice(A270249_gen(),20)) # Chai Wah Wu, Feb 27 2023