%I #20 Feb 27 2023 16:00:39

%S 109,433,2056753,3121201,3577393,26462701,37340353,43823053,128786113,

%T 202705201,304093873,888345793,1005988033,1399680001,1537437133,

%U 2282300173,2310187501,2444964913,2929312513,3564542701,5831255233,7950571201,8512439473,9346947373,9648752833,12627464653,15624660673

%N 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.

%C Subsequence of A243761.

%C 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?

%e 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.

%e 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.

%o (PARI) t(n, p=3) = {while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}

%o for(n=1, 1e3, if(ispseudoprime(P=(3*t(n)^2 + 6*t(n) + 4)) && ispseudoprime(P-2), print1(P, ", ")));

%o (Python)

%o from itertools import islice

%o from sympy import isprime, nextprime

%o def A270249_gen(): # generator of terms

%o p, q = 2, 3

%o while True:

%o if q-p == 2 and isprime(s:=3*p*q+4) and isprime(s-2):

%o yield s

%o p, q = q, nextprime(q)

%o A270249_list = list(islice(A270249_gen(),20)) # _Chai Wah Wu_, Feb 27 2023

%Y Cf. A001359, A243761.

%K nonn

%O 1,1

%A _Altug Alkan_, Mar 14 2016