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A348715
Numbers whose divisors can be partitioned into two disjoint sets without singletons whose harmonic means are both integers.
5
12, 18, 24, 30, 40, 42, 45, 48, 54, 56, 60, 66, 78, 84, 90, 96, 102, 114, 120, 126, 132, 135, 138, 140, 168, 174, 180, 186, 196, 198, 200, 204, 210, 222, 224, 234, 240, 246, 252, 258, 264, 270, 280, 282, 308, 318, 330, 336, 354, 360, 364, 366, 390, 396, 402, 420
OFFSET
1,1
EXAMPLE
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.
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 31 2021
STATUS
approved