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

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, 80, 81, 84, 86, 89, 94, 96, 100, 101, 105, 109, 116, 120, 121, 125, 126, 129, 134, 141, 144, 145, 149, 150, 161, 164, 166, 169, 174, 180, 181, 184, 189, 196, 201, 205, 206, 214, 216

Offset

1,3

Comments

In other words, numbers represented by quadratic form with Gram matrix [1,0; 0,5].

x^2 + 5 y^2 has discriminant -20.

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

References

H. Cohn, A second course in number theory, John Wiley & Sons, Inc., New York-London, 1962. See pp. 3, 4 and later chapters.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989. See Eq. (2.22), p. 33.

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..13859

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Formula

List contains 0 and all positive n such that 2*A035170(n) = A028586(2n) is nonzero. - Michael Somos, Oct 21 2006

Maple

select(t -> [isolve(x^2+5*y^2=t)]<>[], [$0..1000]); # Robert Israel, May 11 2016

Mathematica

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 *)
mx = 300;
limx = Sqrt[mx]; limy = Sqrt[mx/5];
Select[
Union[
Flatten[
Table[x^2 + 5*y^2, {x, 0, limx}, {y, 0, limy}]
       ]
     ], # <= mx &
] (* T. D. Noe, Sep 20 2011 *)

Prog

(Magma) [n: n in [0..216] | NormEquation(5, n) eq true]; // Arkadiusz Wesolowski, May 11 2016

Crossrefs

Cf. A033205, A106865, A154778, A216815, A216816.

For primes see A033205.

For the properly represented numbers see A344231.

Sequence in context: A338692 A039013 A241511 * A091730 A058076 A238712

Adjacent sequences: A020666 A020667 A020668 * A020670 A020671 A020672

Keyword

easy,nonn

Author

David W. Wilson

Extensions

Entry revised by N. J. A. Sloane, Sep 20 2012

Status

approved