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A370813 Number of non-condensed integer factorizations of n into unordered factors > 1.

%I A370813 #7 May 24 2024 03:49:11
%S A370813 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,2,0,0,
%T A370813 0,1,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,4,0,0,0,0,
%U A370813 0,0,0,2,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0
%N A370813 Number of non-condensed integer factorizations of n into unordered factors > 1.
%C A370813 A multiset is condensed iff it is possible to choose a different divisor of each element.
%e A370813 The a(96) = 4 factorizations: (2*2*2*2*2*3), (2*2*2*2*6), (2*2*2*3*4), (2*2*2*12).
%t A370813 facs[n_]:=If[n<=1,{{}},Join @@ Table[Map[Prepend[#,d]&,Select[facs[n/d],Min @@ #>=d&]],{d,Rest[Divisors[n]]}]];
%t A370813 Table[Length[Select[facs[n],Length[Select[Tuples[Divisors /@ #],UnsameQ@@#&]]==0&]],{n,100}]
%Y A370813 Partitions not of this type are counted by A239312, ranks A368110.
%Y A370813 Factors instead of divisors: A368413, complement A368414, unique A370645.
%Y A370813 Partitions of this type are counted by A370320, ranks A355740.
%Y A370813 Subsets of this type: A370583 and A370637, complement A370582 and A370636.
%Y A370813 The complement is counted by A370814, partitions A370592, ranks A368100.
%Y A370813 For a unique choice we have A370815, partitions A370595, ranks A370810.
%Y A370813 A000005 counts divisors.
%Y A370813 A001055 counts factorizations, strict A045778.
%Y A370813 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A370813 Cf. A340596, A340653, A355529, A355739, A355741, A370804, A370807.
%K A370813 nonn
%O A370813 1,32
%A A370813 _Gus Wiseman_, Mar 04 2024