A356765 - OEIS

%I #10 Sep 05 2022 09:10:27

%S 33,35,65,111,209,321,371,395,545,815,1385,1841,1865,4101,5241,6119,

%T 6905,8735,10361,13061,14811,15321,16145,18689,22235,25079,32405,

%U 36095,38789,39395,43739,43829,43881,49745,50811,52331,57701,59195,60035,62765,65561,71931,72329,76019,77135,79751,81311,84395

%N 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.

%H Robert Israel, <a href="/A356765/b356765.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%p filter:= proc(n) local s;

%p   if numtheory:-bigomega(n) <> 2 or issqr(n) then return false fi;

%p   s:= convert( numtheory:-factorset(n),`+`);;

%p   isprime(n+s)

%p   and isprime(n-s)

%p   and isprime(n+2*s) and isprime(n-2*s)

%p end proc:

%p select(filter, [seq(i,i=1..10^5,2)]);

%t 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 *)

%Y Cf. A356762

%K nonn

%O 1,1

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