cp's OEIS Frontend

A370814 Number of condensed integer factorizations of n into unordered factors > 1.

%I A370814 #7 May 24 2024 03:49:36
%S A370814 1,1,1,2,1,2,1,2,2,2,1,4,1,2,2,4,1,4,1,4,2,2,1,6,2,2,2,4,1,5,1,5,2,2,
%T A370814 2,8,1,2,2,6,1,5,1,4,4,2,1,10,2,4,2,4,1,6,2,6,2,2,1,11,1,2,4,7,2,5,1,
%U A370814 4,2,5,1,14,1,2,4,4,2,5,1,10,4,2,1,11,2
%N A370814 Number of condensed integer factorizations of n into unordered factors > 1.
%C A370814 A multiset is condensed iff it is possible to choose a different divisor of each element.
%e A370814 The a(36) = 7 factorizations: (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), (6*6), (36).
%t A370814 facs[n_]:=If[n<=1,{{}},Join @@ Table[Map[Prepend[#,d]&,Select[facs[n/d],Min @@ #>=d&]],{d,Rest[Divisors[n]]}]];
%t A370814 Table[Length[Select[facs[n],Length[Select[Tuples[Divisors /@ #],UnsameQ@@#&]]>0&]],{n,100}]
%Y A370814 Partitions of this type are counted by A239312, ranks A368110.
%Y A370814 Factors instead of divisors: A368414, complement A368413, unique A370645.
%Y A370814 Partitions not of this type are counted by A370320, ranks A355740.
%Y A370814 Subsets of this type: A370582 and A370636, complement A370583 and A370637.
%Y A370814 The complement is counted by A370813, partitions A370593, ranks A355529.
%Y A370814 For a unique choice we have A370815, partitions A370595, ranks A370810.
%Y A370814 A000005 counts divisors.
%Y A370814 A001055 counts factorizations, strict A045778.
%Y A370814 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A370814 Cf. A340596, A340653, A355739, A370592, A370803, A370804, A370805.
%K A370814 nonn
%O A370814 1,4
%A A370814 _Gus Wiseman_, Mar 04 2024