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

1,1

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)

Links

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

Example

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.

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

Prog

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

Crossrefs

Cf. A001358, A173082, A173083

Sequence in context: A147466 A262055 A193276 * A137340 A075640 A147480

Adjacent sequences: A173081 A173082 A173083 * A173085 A173086 A173087

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Feb 09 2010

Extensions

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

Status

approved