A362802 a(n) is the number of ways in which the set of divisors of n can be partitioned into disjoint parts, all of length > 1 and with integer harmonic mean.
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 15, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 175, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 78, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 188, 0
Offset: 1
Keywords
Examples
n a(n) partitions
== ==== ==========
6 1 {{1, 2, 3, 6}}
12 1 {{1, 2, 3, 6}, {4, 12}}
24 4 {{1, 2, 3, 6}, {4, 8, 12, 24}}, {{1, 2, 4, 8, 12, 24}, {3, 6}},
{{1, 3, 6}, {2, 4, 8, 12, 24}}, {{1, 2, 3, 6}, {4, 12}, {8, 24}}
Programs
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Mathematica
harmQ[s_] := AllTrue[s, Length[#] > 1 && IntegerQ[HarmonicMean[#]] &]; a[n_] := Module[{d = Divisors[n], r}, r = ResourceFunction["SetPartitions"][d]; Count[r, _?harmQ]]; Array[a, 119]
Formula
a(A362801(n)) > 0.
Comments