A212707 Semiprimes of the form 5*n^2 + 1.

6, 21, 46, 321, 501, 721, 1126, 2206, 2881, 3646, 3921, 4501, 7606, 10581, 11521, 13521, 14581, 15681, 16246, 18001, 19846, 20481, 21781, 23806, 24501, 27381, 30421, 32001, 38721, 40501, 42321, 48021, 61606, 64981, 72001, 79381, 83206, 89781, 106581, 121681

Offset

1,1

Comments

This is to A137530 (primes of form 1+5n^2) as semiprimes A001358 are to primes A000040. Since Z[sqrt(-5)] is not a unique factorization domain, some numbers of form 1+5n^2 are primes in Z but composite in Z[sqrt(-5)]; some values in this sequence are semiprimes in Z but have a different number than 2 of prime factors in Z[sqrt(-5)].

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

A212656 INTERSECTION A001358.

{k such that 5*n^2 + 1 for a natural number n, and bigomega(k) = A001222(k) = 2}.

Example

a(6) = 721 = 1 + 5*(12^2) = 7 * 103.

Mathematica

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; Select[Table[5*n^2 + 1, {n, 200}], SemiPrimeQ] (* T. D. Noe, May 24 2012 *)
Select[Table[5*n^2 + 1, {n, 180}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

Prog

(Magma) IsSemiprime:= func<n | &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [1..180] | IsSemiprime(s) where s is 5*n^2 + 1]; // Vincenzo Librandi, Sep 22 2012

Crossrefs

Cf. A001222, A001358, A137530, A212656 (5*n^2 + 1).

Sequence in context: A087863 A212656 A051941 * A267370 A213388 A372226

Adjacent sequences: A212704 A212705 A212706 * A212708 A212709 A212710

Keyword

nonn,easy

Author

Jonathan Vos Post, May 24 2012

Extensions

Extended by T. D. Noe, May 24 2012

Status

approved