A173084 Semiprimes q such that q^2-4 and q^2+4 are also semiprimes.

9, 21, 69, 129, 381, 2271, 3849, 3909, 3921, 5001, 5079, 5169, 5349, 7041, 16251, 18129, 18399, 20481, 22569, 22641, 22719, 22809, 28029, 32259, 35151, 38559, 39021, 39441, 39981, 41079, 42459, 48759, 48819, 49431, 50649, 61629, 67929

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Feb 09 2010

Keywords

• nonn

Comments

From Robert Israel, Jun 01 2018: (Start)

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

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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2365

Crossrefs

Cf. A001358, A173082, A173083

Programs

Magma

IsSemiprime:=func; [n: n in [4..7*10^4] | IsSemiprime(n) and IsSemiprime(n^2+4) and IsSemiprime(n^2-4)]; // Vincenzo Librandi, Jun 02 2018

Maple

N:= 10^5: # to get all terms <= N
P:= select(isprime, [seq(i,i=3..N/3,2)]):
select(q -> isprime(q-2) and isprime(q+2) and numtheory:-bigomega(q^2+4)=2, 3*P); # Robert Israel, Jun 01 2018

Mathematica

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

Extensions

Definition reworded by R. J. Mathar, Mar 14 2010