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

1,2

Comments

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

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

Example

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

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 *)

Crossrefs

Sequence in context: A284103 A151674 A367631 * A243000 A285214 A285340

Adjacent sequences: A297965 A297966 A297967 * A297969 A297970 A297971

Keyword

nonn

Author

Robert Israel, Jan 10 2018

Status

approved