A298010 Numbers n such that x*y*(x+y) = n has at least one solution in coprime integers.

2, 6, 12, 20, 30, 42, 56, 70, 72, 84, 90, 110, 120, 126, 132, 156, 180, 182, 198, 210, 240, 264, 272, 286, 306, 308, 330, 342, 380, 390, 420, 462, 468, 506, 510, 520, 546, 552, 600, 624, 630, 646, 650, 660, 702, 714, 756, 798, 812, 840, 870, 880, 884, 912, 930, 966, 992, 1008, 1020, 1056, 1122

Offset: 1

Author

Robert Israel, Jan 10 2018

Keywords

• nonn

Comments

Numbers n such that A297968(n) > 0.

All terms are even.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A297968.

Programs

Maple

filter:= proc(n) local d,x,s,ys;
  d:= numtheory:-divisors(n);
  for x in 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)]);
      if ys <> [] then return true fi;
    fi
  od;
  false
end proc:
select(filter, [seq(i,i=1..10000,2)]);

Mathematica

f[n_] := Module[{d, count, x, s, ys}, d = Divisors[n]; count = 0; Do[If[ IntegerQ[Sqrt[x^4 + 4 n x]], s = Sqrt[x^4 + 4 n x]; ys = Select[{-(s + x^2)/(2x), (x^2 - s)/(2x)}, IntegerQ[#] && GCD[#, x] == 1&]; count = count + Length[ys]], {x, Union[d, -d]}]; count];
Position[Array[f, 2000], ?Positive] // Flatten (* _Jean-François Alcover, Apr 29 2019, after Robert Israel in A297968 *)