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%I A368413 #21 Mar 06 2024 14:47:25
%S A368413 0,0,0,1,0,0,0,2,1,0,0,1,0,0,0,4,0,1,0,1,0,0,0,3,1,0,2,1,0,0,0,6,0,0,
%T A368413 0,4,0,0,0,3,0,0,0,1,1,0,0,7,1,1,0,1,0,3,0,3,0,0,0,2,0,0,1,10,0,0,0,1,
%U A368413 0,0,0,10,0,0,1,1,0,0,0,7,4,0,0,2,0,0
%N A368413 Number of factorizations of n into positive integers > 1 such that it is not possible to choose a different prime factor of each factor.
%C A368413 For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is counted under a(36).
%F A368413 a(n) + A368414(n) = A001055(n).
%e A368413 The a(1) = 0 through a(24) = 3 factorizations:
%e A368413  ... 2*2 ... 2*4   3*3 .. 2*2*3 ... 2*8     . 2*3*3 . 2*2*5 ... 2*2*6
%e A368413              2*2*2                  4*4                         2*3*4
%e A368413                                     2*2*4                       2*2*2*3
%e A368413                                     2*2*2*2
%t A368413 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A368413 Table[Length[Select[facs[n], Select[Tuples[First/@FactorInteger[#]&/@#], UnsameQ@@#&]=={}&]],{n,100}]
%Y A368413 For unlabeled graphs: A140637, complement A134964.
%Y A368413 For labeled graphs: A367867, A367868, A140638, complement A133686.
%Y A368413 For set-systems: A367903, ranks A367907, complement A367902, ranks A367906.
%Y A368413 For non-isomorphic set-systems: A368094, A368409, complement A368095.
%Y A368413 For non-isomorphic multiset partitions: A368097, A355529, A368411.
%Y A368413 Complement for non-isomorphic multiset partitions: A368098, A368100.
%Y A368413 The complement is counted by A368414.
%Y A368413 For non-isomorphic set multipartitions: A368421, complement A368422.
%Y A368413 For divisors instead of prime factors: A370813, complement A370814.
%Y A368413 A001055 counts factorizations, strict A045778.
%Y A368413 A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A368413 A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A368413 A283877 counts non-isomorphic set-systems, connected A300913.
%Y A368413 Cf. A340596, A340653, A367769, A367901, A368187, A368412.
%K A368413 nonn
%O A368413 1,8
%A A368413 _Gus Wiseman_, Dec 27 2023

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