A106594 a(n) = number of primitive solutions to 4n+1 = x^2 + y^2.

1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 2, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 2, 0, 0, 0, 2, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 0, 1, 2, 0, 0, 1, 0, 1

Offset: 1

Author

Colin Mallows, May 10 2005

Keywords

• easy
• nonn

Comments

"Primitive" means that x and y are positive and mutually prime.

Examples

			E.g. a(16)=2 because 65 = 8^2+1^2 = 7^2+4^2. a(11)=0 because although 45=6^2+3^2, 6 and 3 are not mutually prime. a(2)=0 because although 9=3^2+0^2, 0 is not positive.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A106602, A193138.

Programs

Maple

A106594 := proc(n)
      local a,x,y,fourn;
    fourn := 4*n+1 ;
    a := 0 ;
    for x from 1 do
        if x^2 >= fourn then
            return a;
        else
            y := fourn-x^2 ;
            if issqr(y) then
                y := sqrt(y) ;
                if y <= x and igcd(x,y) = 1 then
                    a := a+1 ;
                end if;
            end if;
        end if;
    end do:
end proc: # R. J. Mathar, Sep 21 2013

Mathematica

Table[Length[If[CoprimeQ[#[[1]],#[[2]]],#,Nothing]&/@Union[Sort/@ ({#[[1,2]],#[[2,2]]}&/@FindInstance[{4 n+1==x^2+y^2,x>0,y>0},{x,y}, Integers,10])]],{n,100}] (* Harvey P. Dale, Jun 29 2021 *)