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A154778 Numbers of the form a^2 + 5b^2 with positive integers a,b.

%I A154778 #11 Aug 03 2014 14:01:25
%S A154778 6,9,14,21,24,29,30,36,41,45,46,49,54,56,61,69,70,81,84,86,89,94,96,
%T A154778 101,105,109,116,120,126,129,134,141,144,145,149,150,161,164,166,174,
%U A154778 180,181,184,189,196,201,205,206,214,216,224,225,229,230,241,244,245,246
%N A154778 Numbers of the form a^2 + 5b^2 with positive integers a,b.
%C A154778 Subsequence of A020669 (which allows for a=0 and/or b=0). See there for further references. See A155560 ff for intersection of sequences of type (a^2 + k b^2).
%C A154778 Also, subsequence of A000408 (with 5b^2 = b^2 + (2b)^2).
%e A154778 a(1) = 6 = 1^2 + 5*1^2 is the least number that can be written as A+5B where A,B are positive squares.
%e A154778 a(2) = 9 = 2^2 + 5*1^2 is the second smallest number that can be written in this way.
%t A154778 formQ[n_] := Reduce[a > 0 && b > 0 && n == a^2 + 5 b^2, {a, b}, Integers] =!= False; Select[ Range[300], formQ] (* _Jean-François Alcover_, Sep 20 2011 *)
%t A154778 Timing[mx = 300; limx = Sqrt[mx]; limy = Sqrt[mx/5]; Select[Union[Flatten[Table[x^2 + 5 y^2, {x, limx}, {y, limy}]]], # <= mx &]] (* _T. D. Noe_, Sep 20 2011 *)
%o A154778 (PARI) isA154778(n,/* use optional 2nd arg to get other analogous sequences */c=5) = { for( b=1,sqrtint((n-1)\c), issquare(n-c*b^2) & return(1))}
%o A154778 for( n=1,300, isA154778(n) & print1(n","))
%Y A154778 Cf. A033205 (subsequence of primes). [From _R. J. Mathar_, Jan 26 2009]
%K A154778 easy,nonn
%O A154778 1,1
%A A154778 _M. F. Hasler_, Jan 24 2009