A297968 Number of solutions to x*y*(x+y)=n in coprime integers.

0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Robert Israel, Jan 10 2018

Keywords

• nonn

Comments

a(n)=0 if n is odd. - Robert Israel, Jan 10 2018

Examples

			For n=6 the a(n)=6 solutions are (x,y) = (-3,1), (-3,2), (1,-3), (1,2), (2,1) and (2,-3).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• C. L. Stewart, On the number of solutions of polynomial congruences and Thue equations, J. Amer. Math. Soc. 4 (1991), 793-835.
• S. Y. Xiao et al, Integers h such that xy(x+y)=h has many integer solutions, Math Overflow

Programs

Maple

f:= proc(n) local d,count,x,s,ys;
  d:= numtheory:-divisors(n);
  count:= 0:
  for x in d union map(`-`,d) do
    if issqr(x^4+4*n*x) then
      s:= sqrt(x^4+4*n*x);
      ys:= select(t -> type(t,integer) and igcd(t,x)=1, [-(s+x^2)/(2*x), (x^2-s)/(2*x)]);
      count:= count + nops(ys);
    fi
  od;
  count
end proc:
map(f, [$1..200]);

Mathematica

f[n_] := Module[{d, count, x, s, ys}, d = Divisors[n]; count = 0; Do[If[ IntegerQ[Sqrt[x^4 + 4n x]], s = Sqrt[x^4 + 4n x]; ys = Select[{-(s+x^2)/ (2x), (x^2-s)/(2x)}, IntegerQ[#] && GCD[#, x] == 1&]; count = count + Length[ys]], {x, Union[d, -d]}]; count]; Array[f, 200] (* Jean-François Alcover, Apr 29 2019, after Robert Israel *)