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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.

%I A358393 #12 Nov 21 2022 09:49:45
%S A358393 261977,496163,1943101,2204273,2502827,2632627,2822381,2878543,
%T A358393 3291593,3431891,4122043,4269679,5205671,5224361,5565139,6248881,
%U A358393 6600989,6881291,7568963,8181317,8251277,8377777,9005561,9644911,10226233,11096753,11767801,12252271,13197361,13574489,13730263,14064901
%N 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.
%H A358393 Robert Israel, <a href="/A358393/b358393.txt">Table of n, a(n) for n = 1..2000</a>
%e A358393 a(1) = 261977 is a term because 261977, 261983 and 262007 are consecutive primes with 261977*261983 + 261977*262007 - 261983*262007 = 68631948349,
%e A358393 261977*261983 - 261977*262007 + 261983*262007 = 68635092433, and
%e A358393 -261977*261983 + 261977*262007 + 261983*262007 = 68647667329 prime.
%p A358393 q:= 2: r:= 3:
%p A358393 R:= NULL: count:= 0:
%p A358393 while count < 40 do
%p A358393   p:= q; q:= r; r:= nextprime(r);
%p A358393   s:= p*(q+r)+q*r;
%p A358393   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;
%p A358393   fi
%p A358393 od:
%p A358393 R;
%t A358393 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 *)
%o A358393 (Python)
%o A358393 from itertools import islice
%o A358393 from sympy import isprime, nextprime
%o A358393 def agen():
%o A358393     p, q, r = 2, 3, 5
%o A358393     while True:
%o A358393         pq, pr, qr = p*q, p*r, q*r
%o A358393         if all(isprime(t) for t in [pq+pr-qr, pq-pr+qr, -pq+pr+qr]): yield p
%o A358393         p, q, r = q, r, nextprime(r)
%o A358393 print(list(islice(agen(), 15))) # _Michael S. Branicky_, Nov 13 2022
%Y A358393 Contained in A054643.
%K A358393 nonn
%O A358393 1,1
%A A358393 _J. M. Bergot_ and _Robert Israel_, Nov 13 2022