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

%I A298010 #12 Apr 29 2019 12:25:52
%S A298010 2,6,12,20,30,42,56,70,72,84,90,110,120,126,132,156,180,182,198,210,
%T A298010 240,264,272,286,306,308,330,342,380,390,420,462,468,506,510,520,546,
%U A298010 552,600,624,630,646,650,660,702,714,756,798,812,840,870,880,884,912,930,966,992,1008,1020,1056,1122
%N A298010 Numbers n such that x*y*(x+y) = n has at least one solution in coprime integers.
%C A298010 Numbers n such that A297968(n) > 0.
%C A298010 All terms are even.
%H A298010 Robert Israel, <a href="/A298010/b298010.txt">Table of n, a(n) for n = 1..10000</a>
%p A298010 filter:= proc(n) local d,x,s,ys;
%p A298010   d:= numtheory:-divisors(n);
%p A298010   for x in d do
%p A298010     if issqr(x^4+4*n*x) then
%p A298010       s:= sqrt(x^4+4*n*x);
%p A298010       ys:= select(t -> type(t,integer) and igcd(t,x)=1, [-(s+x^2)/(2*x), (x^2-s)/(2*x)]);
%p A298010       if ys <> [] then return true fi;
%p A298010     fi
%p A298010   od;
%p A298010   false
%p A298010 end proc:
%p A298010 select(filter, [seq(i,i=1..10000,2)]);
%t A298010 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];
%t A298010 Position[Array[f, 2000], _?Positive] // Flatten (* _Jean-François Alcover_, Apr 29 2019, after _Robert Israel_ in A297968 *)
%Y A298010 Cf. A297968.
%K A298010 nonn
%O A298010 1,1
%A A298010 _Robert Israel_, Jan 10 2018