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Primes p such that p^2 - p*q + q^2 is prime, where q is the next prime after p.
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%I #17 Feb 11 2021 03:03:00

%S 2,3,53,151,167,263,373,443,467,509,523,571,1063,1103,1117,1217,1493,

%T 1553,1901,1973,2161,2207,2281,2399,2713,2837,2963,3259,3347,3511,

%U 4073,4297,4373,4463,4523,4673,4691,4877,5147,5237,5303,5419,5471,5851,6211,6311,6367

%N Primes p such that p^2 - p*q + q^2 is prime, where q is the next prime after p.

%C From _Bernard Schott_, Dec 23 2020: (Start)

%C Except for a(2)=3, (3, 5) gives A339698(2) = 19, there is no other pair of twin primes (p, p+2) (p in A001359) that gives a prime number of the form p^2-p*q+q^2 = p^2+2p+4.

%C There are no consecutive cousin primes (p, p+4) (p in A029710) that gives a prime number of the form p^2-pq+q^2 = p^2+4p+16.

%C There are no consecutive primes with a gap of 8 (p, p+8) (p in A031926) that give a prime number of the form p^2-pq+q^2 = p^2+8p+64. (End)

%H Robert Israel, <a href="/A339920/b339920.txt">Table of n, a(n) for n = 1..10000</a>

%p q:= 2: count:= 0: R:= NULL:

%p while count < 100 do

%p p:= q; q:= nextprime(q);

%p if isprime(p^2-p*q+q^2) then

%p count:= count+1; R:= R, p;

%p fi

%p od:

%p R; # _Robert Israel_, Dec 24 2020

%o (PARI) forprime(p=1, 1e4, my(q=nextprime(p+1)); if(ispseudoprime(p^2-p*q+q^2), print1(p, ", ")));

%Y Cf. A339698.

%K nonn

%O 1,1

%A _Michel Marcus_, Dec 23 2020