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Primes of the form (p^2 - p*q + q^2)/3, where p and q are consecutive primes.
2

%I #12 Dec 31 2023 17:23:36

%S 13,31,79,109,151,1201,3271,3469,3889,4111,12289,16879,17791,25951,

%T 27673,108301,126079,134857,138679,169957,174259,186019,231877,245389,

%U 259309,355009,367501,371737,397489,412939,461017,477619,524197,544429,565069,602401,741031,833191,904303,961069,1267501

%N Primes of the form (p^2 - p*q + q^2)/3, where p and q are consecutive primes.

%H Robert Israel, <a href="/A342706/b342706.txt">Table of n, a(n) for n = 1..10000</a>

%e For n = 5, p = 19 and q = 23 are consecutive primes and a(5) = (19^2-19*23+23^2)/3 = 151 is prime.

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

%p while count<100 do

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

%p r:= (p^2-p*q+q^2)/3;

%p if r::integer and isprime(r) then

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

%p fi;

%p od:

%p R;

%t cpQ[{a_,b_}]:=Module[{c=(a^2-a*b+b^2)/3},If[PrimeQ[c],c,Nothing]]; cpQ/@Partition[Prime[ Range[ 500]],2,1] (* _Harvey P. Dale_, Dec 31 2023 *)

%o (Python)

%o from sympy import isprime, nextprime

%o def aupto(limit):

%o p, q, num, alst = 3, 5, 7, []

%o while num//3 <= limit:

%o if num%3 == 0 and isprime(num//3): alst.append(num//3)

%o p, q, num = q, nextprime(q), p**2 - p*q + q**2

%o return alst

%o print(aupto(1267501)) # _Michael S. Branicky_, Mar 18 2021

%Y Cf. A339698, A342705.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Mar 18 2021