login
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.
2

%I #9 May 05 2023 01:41:30

%S 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,

%T 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,

%U 0,0,0,0,78,0,0,0,0,0,1,0,0,0,0,0,188,0

%N 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.

%F a(A362801(n)) > 0.

%e n a(n) partitions

%e == ==== ==========

%e 6 1 {{1, 2, 3, 6}}

%e 12 1 {{1, 2, 3, 6}, {4, 12}}

%e 24 4 {{1, 2, 3, 6}, {4, 8, 12, 24}}, {{1, 2, 4, 8, 12, 24}, {3, 6}},

%e {{1, 3, 6}, {2, 4, 8, 12, 24}}, {{1, 2, 3, 6}, {4, 12}, {8, 24}}

%t 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]

%Y Cf. A339453, A339665, A362801, A362803 (indices of records).

%K nonn

%O 1,24

%A _Amiram Eldar_, May 04 2023