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

Author

David W. Wilson

Keywords

• easy
• nonn

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)

Crossrefs

Cf. A033205, A106865, A154778, A216815, A122870.

For primes see A033205.

For the properly represented numbers see A344231.

Programs

Magma

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

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 *)

Formula

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

Extensions

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