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

1,1

Comments

Numbers n such that A297968(n) > 0.

All terms are even.

Links

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

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 *)

Crossrefs

Cf. A297968.

Sequence in context: A254949 A034100 A034110 * A118373 A077252 A102069

Adjacent sequences: A298007 A298008 A298009 * A298011 A298012 A298013

Keyword

nonn

Author

Robert Israel, Jan 10 2018

Status

approved