A173084 - OEIS

%I #15 Aug 29 2024 18:30:29

%S 9,21,69,129,381,2271,3849,3909,3921,5001,5079,5169,5349,7041,16251,

%T 18129,18399,20481,22569,22641,22719,22809,28029,32259,35151,38559,

%U 39021,39441,39981,41079,42459,48759,48819,49431,50649,61629,67929

%N Semiprimes q such that q^2-4 and q^2+4 are also semiprimes.

%C From _Robert Israel_, Jun 01 2018: (Start)

%C Since q^2-4 = (q-2)(q+2), for this to be a semiprime requires q-2 and q+2 to be primes.

%C All terms == 3 (mod 6), thus q/3 is an odd prime. (End)

%H Robert Israel, <a href="/A173084/b173084.txt">Table of n, a(n) for n = 1..2365</a>

%e 9^2-4 = 77 = 7*11 and 9^2+4 = 85 = 5*13 are semiprimes created by q=9, which adds the semiprime q=9 to the sequence.

%p N:= 10^5: # to get all terms <= N

%p P:= select(isprime, [seq(i,i=3..N/3,2)]):

%p select(q -> isprime(q-2) and isprime(q+2) and numtheory:-bigomega(q^2+4)=2, 3*P); # _Robert Israel_, Jun 01 2018

%t f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2}; lst={}; Do[If[f[n], a=n^2-4;b=n^2+4;If[f[a]&&f[b],AppendTo[lst,n]]],{n,9!}]; lst

%o (Magma) IsSemiprime:=func<n | &+[d[2]: d in Factorization(n)] eq 2>; [n: n in [4..7*10^4] | IsSemiprime(n) and IsSemiprime(n^2+4) and IsSemiprime(n^2-4)]; // _Vincenzo Librandi_, Jun 02 2018

%Y Cf. A001358, A173082, A173083

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Feb 09 2010

%E Definition reworded by _R. J. Mathar_, Mar 14 2010