A339698 -id:A339698 - cp's OEIS frontend

A339920 Primes p such that p^2 - p*q + q^2 is prime, where q is the next prime after p.

2, 3, 53, 151, 167, 263, 373, 443, 467, 509, 523, 571, 1063, 1103, 1117, 1217, 1493, 1553, 1901, 1973, 2161, 2207, 2281, 2399, 2713, 2837, 2963, 3259, 3347, 3511, 4073, 4297, 4373, 4463, 4523, 4673, 4691, 4877, 5147, 5237, 5303, 5419, 5471, 5851, 6211, 6311, 6367

Offset: 1

Michel Marcus, Dec 23 2020

nonn

From Bernard Schott, Dec 23 2020: (Start)
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.
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.
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)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A339698.

q:= 2: count:= 0: R:= NULL:
while count < 100 do
  p:= q; q:= nextprime(q);
  if isprime(p^2-p*q+q^2) then
    count:= count+1; R:= R, p;
  fi
od:
R; # Robert Israel, Dec 24 2020

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

A342706 Primes of the form (p^2 - p*q + q^2)/3, where p and q are consecutive primes.

Original entry on oeis.org

13, 31, 79, 109, 151, 1201, 3271, 3469, 3889, 4111, 12289, 16879, 17791, 25951, 27673, 108301, 126079, 134857, 138679, 169957, 174259, 186019, 231877, 245389, 259309, 355009, 367501, 371737, 397489, 412939, 461017, 477619, 524197, 544429, 565069, 602401, 741031, 833191, 904303, 961069, 1267501

Offset: 1

J. M. Bergot and Robert Israel, Mar 18 2021

nonn

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

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A339698, A342705.

R:= NULL: q:= 2: count:= 0:
while count<100 do
  p:= q; q:= nextprime(p);
  r:= (p^2-p*q+q^2)/3;
  if r::integer and isprime(r) then
    count:= count+1; R:= R, r;
  fi;
od:
R;

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 *)

from sympy import isprime, nextprime
def aupto(limit):
  p, q, num, alst = 3, 5, 7, []
  while num//3 <= limit:
    if num%3 == 0 and isprime(num//3): alst.append(num//3)
    p, q, num = q, nextprime(q), p**2 - p*q + q**2
  return alst
print(aupto(1267501)) # Michael S. Branicky, Mar 18 2021

cp's OEIS Frontend

A339920 Primes p such that p^2 - p*q + q^2 is prime, where q is the next prime after p.

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