|
|
A173084
|
|
Semiprimes q such that q^2-4 and q^2+4 are also semiprimes.
|
|
4
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
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
|
|
|
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
|
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|