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A297968 Number of solutions to x*y*(x+y)=n in coprime integers. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n)=0 if n is odd. - Robert Israel, Jan 10 2018
LINKS
C. L. Stewart, On the number of solutions of polynomial congruences and Thue equations, J. Amer. Math. Soc. 4 (1991), 793-835.
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
KEYWORD
nonn
AUTHOR
Robert Israel, Jan 10 2018
STATUS
approved

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Last modified June 16 08:45 EDT 2024. Contains 373424 sequences. (Running on oeis4.)