A358393 First of three consecutive primes p,q,r such that p*q + p*r - q*r, p*q - p*r + q*r and -p*q + p*r + q*r are all prime.

261977, 496163, 1943101, 2204273, 2502827, 2632627, 2822381, 2878543, 3291593, 3431891, 4122043, 4269679, 5205671, 5224361, 5565139, 6248881, 6600989, 6881291, 7568963, 8181317, 8251277, 8377777, 9005561, 9644911, 10226233, 11096753, 11767801, 12252271, 13197361, 13574489, 13730263, 14064901

Offset: 1

Author

J. M. Bergot and Robert Israel, Nov 13 2022

Keywords

• nonn

Examples

			a(1) = 261977 is a term because 261977, 261983 and 262007 are consecutive primes with 261977*261983 + 261977*262007 - 261983*262007 = 68631948349,
261977*261983 - 261977*262007 + 261983*262007 = 68635092433, and
-261977*261983 + 261977*262007 + 261983*262007 = 68647667329 prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2000

Crossrefs

Contained in A054643.

Programs

Maple

q:= 2: r:= 3:
R:= NULL: count:= 0:
while count < 40 do
  p:= q; q:= r; r:= nextprime(r);
  s:= p*(q+r)+q*r;
  if  isprime(s-2*p*q) and isprime(s-2*p*r) and isprime(s-2*q*r) then       R:= R, p; count:= count+1;
  fi
od:
R;

Mathematica

f[p_, q_, r_] := PrimeQ[p*q + p*r - q*r] && PrimeQ[p*q - p*r + q*r] && PrimeQ[-p*q + p*r + q*r]; Select[Partition[Prime[Range[10^6]], 3, 1], f @@ # &][[;; , 1]] (* Amiram Eldar, Nov 13 2022 *)

Python

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p, q, r = 2, 3, 5
    while True:
        pq, pr, qr = p*q, p*r, q*r
        if all(isprime(t) for t in [pq+pr-qr, pq-pr+qr, -pq+pr+qr]): yield p
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 15))) # Michael S. Branicky, Nov 13 2022