cp's OEIS Frontend

A020669 Numbers of form x^2 + 5 y^2.

%I A020669 #70 Jun 22 2025 17:33:37
%S A020669 0,1,4,5,6,9,14,16,20,21,24,25,29,30,36,41,45,46,49,54,56,61,64,69,70,
%T A020669 80,81,84,86,89,94,96,100,101,105,109,116,120,121,125,126,129,134,141,
%U A020669 144,145,149,150,161,164,166,169,174,180,181,184,189,196,201,205,206,214,216
%N A020669 Numbers of form x^2 + 5 y^2.
%C A020669 In other words, numbers represented by quadratic form with Gram matrix [1,0; 0,5].
%C A020669 x^2 + 5 y^2 has discriminant -20.
%C A020669 A positive integer n is in this sequence if and only if the p-adic order ord_p(n) of n is even for any prime p with floor(p/10) odd, and the number of prime divisors p == 3 or 7 (mod 20) of n with ord_p(n) odd has the same parity with ord_2(n). - _Zhi-Wei Sun_, Mar 24 2018
%D A020669 H. Cohn, A second course in number theory, John Wiley & Sons, Inc., New York-London, 1962. See pp. 3, 4 and later chapters.
%D A020669 David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989. See Eq. (2.22), p. 33.
%H A020669 N. J. A. Sloane, <a href="/A020669/b020669.txt">Table of n, a(n) for n = 1..13859</a>
%H A020669 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%F A020669 List contains 0 and all positive n such that 2*A035170(n) = A028586(2n) is nonzero. - _Michael Somos_, Oct 21 2006
%p A020669 select(t -> [isolve(x^2+5*y^2=t)]<>[], [$0..1000]); # _Robert Israel_, May 11 2016
%t A020669 formQ[n_] := Reduce[x >= 0 && y >= 0 && n == x^2 + 5 y^2, {x, y}, Integers] =!= False; Select[ Range[0, 300], formQ] (* _Jean-François Alcover_, Sep 20 2011 *)
%t A020669 mx = 300;
%t A020669 limx = Sqrt[mx]; limy = Sqrt[mx/5];
%t A020669 Select[
%t A020669 Union[
%t A020669 Flatten[
%t A020669 Table[x^2 + 5*y^2, {x, 0, limx}, {y, 0, limy}]
%t A020669        ]
%t A020669      ], # <= mx &
%t A020669 ] (* _T. D. Noe_, Sep 20 2011 *)
%o A020669 (Magma) [n: n in [0..216] | NormEquation(5, n) eq true]; // _Arkadiusz Wesolowski_, May 11 2016
%Y A020669 Cf. A033205, A106865, A154778, A216815, A122870.
%Y A020669 For primes see A033205.
%Y A020669 For the properly represented numbers see A344231.
%K A020669 easy,nonn
%O A020669 1,3
%A A020669 _David W. Wilson_
%E A020669 Entry revised by _N. J. A. Sloane_, Sep 20 2012