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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

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]);

CROSSREFS

Sequence in context: A071326 A284103 A151674 * A243000 A285214 A285340

Adjacent sequences:  A297965 A297966 A297967 * A297969 A297970 A297971

KEYWORD

nonn

AUTHOR

Robert Israel, Jan 10 2018

STATUS

approved

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Last modified October 24 01:20 EDT 2018. Contains 316541 sequences. (Running on oeis4.)