|
%I #14 Sep 08 2022 08:46:02
%S 6,21,46,321,501,721,1126,2206,2881,3646,3921,4501,7606,10581,11521,
%T 13521,14581,15681,16246,18001,19846,20481,21781,23806,24501,27381,
%U 30421,32001,38721,40501,42321,48021,61606,64981,72001,79381,83206,89781,106581,121681
%N Semiprimes of the form 5*n^2 + 1.
%C 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)].
%H Vincenzo Librandi, <a href="/A212707/b212707.txt">Table of n, a(n) for n = 1..1000</a>
%F A212656 INTERSECTION A001358.
%F {k such that 5*n^2 + 1 for a natural number n, and bigomega(k) = A001222(k) = 2}.
%e a(6) = 721 = 1 + 5*(12^2) = 7 * 103.
%t 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 *)
%t Select[Table[5*n^2 + 1, {n, 180}], PrimeOmega[#] == 2&] (* _Vincenzo Librandi_, Sep 22 2012 *)
%o (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
%Y Cf. A001222, A001358, A137530, A212656 (5*n^2 + 1).
%K nonn,easy
%O 1,1
%A _Jonathan Vos Post_, May 24 2012
%E Extended by _T. D. Noe_, May 24 2012
|
