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%I #5 Oct 31 2021 09:35:43
%S 12,18,24,30,40,42,45,48,54,56,60,66,78,84,90,96,102,114,120,126,132,
%T 135,138,140,168,174,180,186,196,198,200,204,210,222,224,234,240,246,
%U 252,258,264,270,280,282,308,318,330,336,354,360,364,366,390,396,402,420
%N Numbers whose divisors can be partitioned into two disjoint sets without singletons whose harmonic means are both integers.
%e 12 is a term since its set of divisors, {1, 2, 3, 4, 6, 12}, can be partitioned into the two disjoint sets, {1, 2, 3, 6} and {4, 12}, whose harmonic means, 2 and 6 respectively, are both integers.
%t hQ[d_] := IntegerQ @ HarmonicMean[d]; q[n_] := Module[{d = Divisors[n], nd, s, subs, ans = False}, nd = Length[d]; subs = Subsets[d]; Do[s = subs[[k]]; If[Length[s] > 1 && Length[s] <= nd/2 && hQ[s] && hQ[Complement[d, s]], ans = True; Break[]], {k, 1, Length[subs]}]; ans]; Select[Range[300], q]
%Y Cf. A001599, A027750, A083207.
%K nonn
%O 1,1
%A _Amiram Eldar_, Oct 31 2021