%I #24 Jun 12 2024 15:32:25

%S 5,29,599,26699,59669,72869,189389,285839,389999,508619,623669,708989,

%T 862229,908879,945629,945809,953789,1002149,1134389,1138409,1431569,

%U 1461209,1712549,2110289,2127269,2158589,2704769,2727299,2837279,3004049,3068909,3091379,3280229,3336659,3402239,3546269

%N Primes p such that p+2, (p^2-5)/2-p, (p^2-1)/2+p, and (p^2+3)/2+3*p are all prime.

%C Lower twin primes p such that if q = p+2, then (p*q-1)/2, (p*q-1)/2-p-q and (p*q-1)/2+p+q are also prime.

%C All terms but the first == 29 (mod 30).

%H Robert Israel, <a href="/A352951/b352951.txt">Table of n, a(n) for n = 1..2500</a>

%e a(3)=599 is a term because it, 599+2 = 601, (599*601-1)/2 = 179999, 179999-599-601 = 178799, and 179999+599+601 = 181199 are prime.

%p R:= 5: count:= 0:

%p for p from 29 by 30 while count < 60 do

%p if isprime(p) and isprime(p+2) then

%p q:= p+2; r:= (p*q-1)/2;

%p if isprime(r) and isprime(r+p+q) and isprime(r-p-q) then

%p count:= count+1; R:= R,p;

%p fi

%p fi

%p od:

%p R;

%t Select[Prime[Range[250000]], And @@ PrimeQ[{# + 2, (#^2 - 5)/2 - #, (#^2 - 1)/2 + #, (#^2 + 3)/2 + 3*#}] &] (* _Amiram Eldar_, Apr 11 2022 *)

%t Select[Prime[Range[260000]],AllTrue[{#+2,(#^2-5)/2-#,(#^2-1)/2+#,(#^2+3)/2+3#},PrimeQ]&] (* _Harvey P. Dale_, Jun 12 2024 *)

%o (Python)

%o from itertools import islice

%o from sympy import isprime, nextprime

%o def agen(): # generator of terms

%o p, q = 3, 5

%o while True:

%o if q == p+2:

%o t, s = (p*q-1)//2, p+q

%o if isprime(t) and isprime(t+s) and isprime(t-s):

%o yield p

%o p, q = q, nextprime(q)

%o print(list(islice(agen(), 36))) # _Michael S. Branicky_, Apr 10 2022

%Y Cf. A352948.

%Y Subsequence of A001359.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Apr 10 2022