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

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

1,1

Links

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

Example

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

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, r;
  fi;
od:
R;

Mathematica

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 *)

Prog

(Python)
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

Crossrefs

Cf. A339698, A342705.

Sequence in context: A211116 A107288 A335732 * A095379 A350095 A160772

Adjacent sequences: A342703 A342704 A342705 * A342707 A342708 A342709

Keyword

nonn

Author

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

Status

approved