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%I A370645 #8 Mar 03 2024 21:01:39
%S A370645 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,
%T A370645 1,4,1,1,1,3,1,1,1,2,2,1,1,4,1,2,1,2,1,3,1,3,1,1,1,3,1,1,2,1,1,1,1,2,
%U A370645 1,1,1,5,1,1,2,2,1,1,1,4,1,1,1,3,1,1,1
%N A370645 Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.
%C A370645 All of these factorizations are co-balanced (A340596).
%e A370645 The factorization f = (3*6*10) has prime factor choices (3,2,2), (3,3,2), (3,2,5), and (3,3,5), of which only (3,2,5) has all different parts, so f is counted under a(180).
%e A370645 The a(n) factorizations for n = 2, 12, 24, 36, 72, 120, 144, 180, 288:
%e A370645   (2)  (2*6)  (3*8)   (4*9)   (8*9)   (3*5*8)   (2*72)   (4*5*9)   (3*96)
%e A370645        (3*4)  (4*6)   (6*6)   (2*36)  (4*5*6)   (3*48)   (5*6*6)   (4*72)
%e A370645               (2*12)  (2*18)  (3*24)  (2*3*20)  (4*36)   (2*3*30)  (6*48)
%e A370645                       (3*12)  (4*18)  (2*5*12)  (6*24)   (2*5*18)  (8*36)
%e A370645                               (6*12)  (2*6*10)  (8*18)   (2*6*15)  (9*32)
%e A370645                                       (3*4*10)  (9*16)   (2*9*10)  (12*24)
%e A370645                                                 (12*12)  (3*4*15)  (16*18)
%e A370645                                                          (3*5*12)  (2*144)
%e A370645                                                          (3*6*10)
%t A370645 facs[n_]:=If[n<=1,{{}},Join @@ Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
%t A370645 Table[Length[Select[facs[n], Length[Union[Sort/@Select[Tuples[First /@ FactorInteger[#]&/@#], UnsameQ@@#&]]]==1&]],{n,100}]
%Y A370645 Multisets of this type are ranked by A368101, see also A368100, A355529.
%Y A370645 For nonexistence we have A368413, complement A368414.
%Y A370645 Subsets of this type are counted by A370584, see also A370582, A370583.
%Y A370645 Maximal sets of this type are counted by A370585.
%Y A370645 The version for partitions is A370594, see also A370592, A370593.
%Y A370645 Subsets of this type are counted by A370638, see also A370636, A370637.
%Y A370645 For unlabeled multiset partitions we have A370646, also A368098, A368097.
%Y A370645 A001055 counts factorizations, strict A045778.

%Y A370645 A006530 gives greatest prime factor, least A020639.
%Y A370645 A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A370645 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370645 A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A370645 A283877 counts non-isomorphic set-systems, connected A300913.
%Y A370645 A355741 counts ways to choose a prime factor of each prime index.
%Y A370645 For set-systems see A367902-A367908.
%Y A370645 Cf. A000040, A000720, A003963, A340596, A340653, A355744, A355745, A368110.
%K A370645 nonn
%O A370645 1,12
%A A370645 _Gus Wiseman_, Mar 01 2024

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