A356765 Semiprimes p*q such that p*q+p+q, p*q-(p+q), p*q+2*(p+q) and p*q-2*(p+q) are all primes.

33, 35, 65, 111, 209, 321, 371, 395, 545, 815, 1385, 1841, 1865, 4101, 5241, 6119, 6905, 8735, 10361, 13061, 14811, 15321, 16145, 18689, 22235, 25079, 32405, 36095, 38789, 39395, 43739, 43829, 43881, 49745, 50811, 52331, 57701, 59195, 60035, 62765, 65561, 71931, 72329, 76019, 77135, 79751, 81311, 84395

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(3) = 65 = 5*13 is a term because 5*13+5+13 = 83, 5*13-(5+13) = 47, 5*13+2*(5+13) = 101 and 5*13-2*(5+13) = 29 are all prime.

Maple

filter:= proc(n) local s;
  if numtheory:-bigomega(n) <> 2 or issqr(n) then return false fi;
  s:= convert( numtheory:-factorset(n), `+`);;
  isprime(n+s)
  and isprime(n-s)
  and isprime(n+2*s) and isprime(n-2*s)
end proc:
select(filter, [seq(i, i=1..10^5, 2)]);

Mathematica

Select[Range[10^5], (f = FactorInteger[#])[[;; , 2]] == {1, 1} && AllTrue[{(p = f[[1, 1]])*(q = f[[2, 1]]) + p + q, p*q - (p + q), p*q + 2*(p + q), p*q - 2*(p + q)}, PrimeQ] &] (* Amiram Eldar, Aug 26 2022 *)

Crossrefs

Cf. A356762

Sequence in context: A144425 A180329 A367782 * A020260 A345500 A345501

Adjacent sequences: A356762 A356763 A356764 * A356766 A356767 A356768

Keyword

nonn

Author

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

Status

approved