A356762 - OEIS

%I #16 Sep 05 2022 09:10:37

%S 5,50929,74759,127541,349849,1287731,1294753,3941711,4190023,6130739,

%T 6310061,6593329,6816973,7347709,7573849,8690351,9813409,10985959,

%U 11703187,12130553,12504001,18032059,18468763,20207471,21357191,23635603,24301309,25078181,28509521,28729567,28855459,30200411,31304239

%N Primes p such that, if q is the next prime, p*q+p+q, p*q-p-q, p*q+2*(p+q) and p*q-2*(p+q) are all prime.

%H Robert Israel, <a href="/A356762/b356762.txt">Table of n, a(n) for n = 1..1000</a>

%e a(3) = 74759 is a term because it is prime, the next prime is 74761, and

%e 74759*74761 + 74759 + 74761 = 5589207119

%e 74759*74761 - 74759 - 74761 = 5588908079

%e 74759*74761 + 2*(74759 + 74761) = 5589356639

%e 74759*74761 - 2*(74759 + 74761) = 5588758559

%e are all prime.

%p q:= 2: R:= NULL: count:= 0:

%p while count < 40 do

%p   p:= q; q:= nextprime(q); if isprime(p*q+p+q) and isprime(p*q-p-q) and isprime(p*q+2*p+2*q) and

%p isprime(p*q-2*p-2*q) then count:= count+1; R:= R,p; fi

%p od:

%p R;

%t Select[Partition[Prime[Range[2*10^6]], 2, 1], AllTrue[{(p = #[[1]])*(q = #[[2]]) + p + q, p*q - p - q, p*q + 2*(p + q), p*q - 2*(p + q)}, PrimeQ] &][[;; , 1]] (* _Amiram Eldar_, Aug 26 2022 *)

%o (Python)

%o from sympy import isprime, nextprime

%o from itertools import count, islice

%o def agen(): # generator of terms

%o     p, q = 2, 3

%o     while True:

%o         if all(isprime(t) for t in [p*q+p+q, p*q-p-q, p*q+2*(p+q), p*q-2*(p+q)]):

%o             yield p

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

%o print(list(islice(agen(), 15))) # _Michael S. Branicky_, Aug 26 2022

%Y Cf. A356765.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Aug 26 2022