A356762 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.

5, 50929, 74759, 127541, 349849, 1287731, 1294753, 3941711, 4190023, 6130739, 6310061, 6593329, 6816973, 7347709, 7573849, 8690351, 9813409, 10985959, 11703187, 12130553, 12504001, 18032059, 18468763, 20207471, 21357191, 23635603, 24301309, 25078181, 28509521, 28729567, 28855459, 30200411, 31304239

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Example

a(3) = 74759 is a term because it is prime, the next prime is 74761, and
74759*74761 + 74759 + 74761 = 5589207119
74759*74761 - 74759 - 74761 = 5588908079
74759*74761 + 2*(74759 + 74761) = 5589356639
74759*74761 - 2*(74759 + 74761) = 5588758559
are all prime.

Maple

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

Mathematica

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 *)

Prog

(Python)
from sympy import isprime, nextprime
from itertools import count, islice
def agen(): # generator of terms
    p, q = 2, 3
    while True:
        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)]):
            yield p
        p, q = q, nextprime(q)
print(list(islice(agen(), 15))) # Michael S. Branicky, Aug 26 2022

Crossrefs

Cf. A356765.

Sequence in context: A165711 A167369 A259161 * A242833 A242478 A247845

Adjacent sequences: A356759 A356760 A356761 * A356763 A356764 A356765

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Aug 26 2022

Status

approved