A342705 Primes p such that (p^2 - p*q + q^2)/3 is prime, where q is the next prime after p.

5, 7, 13, 17, 19, 59, 97, 101, 107, 109, 191, 223, 229, 277, 283, 569, 613, 631, 643, 709, 719, 743, 829, 857, 881, 1031, 1049, 1051, 1091, 1109, 1171, 1193, 1249, 1277, 1301, 1327, 1489, 1579, 1637, 1697, 1949, 1979, 2003, 2081, 2089, 2113, 2141, 2203, 2357, 2423, 2539, 2593, 2659, 2749, 2789, 2819

Offset: 1

Author

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

Keywords

• nonn

Comments

If p == -q (mod 3) then p^2 - p*q + q^2 is divisible by 3.

Examples

			a(5) = 19 is a term because 19 and 23 are consecutive primes and (19^2 - 19*23 + 23^2)/3 = 151 is prime.
		

Links

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

Crossrefs

Cf. A339920, A342706.

Programs

Maple

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, p;
  fi;
od:
R;

Python

from sympy import isprime, nextprime
def aupto(limit):
  p, q, num, alst = 2, 3, 7, []
  while  p <= limit:
    if num%3 == 0 and isprime(num//3): alst.append(p)
    p, q = q, nextprime(q)
    num = p**2 - p*q + q**2
  return alst
print(aupto(2819)) # Michael S. Branicky, Mar 18 2021