cp's OEIS Frontend

A368414 Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each factor.

%I A368414 #15 Mar 06 2024 14:47:48
%S A368414 1,1,1,1,1,2,1,1,1,2,1,3,1,2,2,1,1,3,1,3,2,2,1,4,1,2,1,3,1,5,1,1,2,2,
%T A368414 2,5,1,2,2,4,1,5,1,3,3,2,1,5,1,3,2,3,1,4,2,4,2,2,1,9,1,2,3,1,2,5,1,3,
%U A368414 2,5,1,6,1,2,3,3,2,5,1,5,1,2,1,9,2,2,2
%N A368414 Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each factor.
%C A368414 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 not counted under a(36).
%F A368414 a(n) = A001055(n) - A368413(n).
%e A368414 The a(n) factorizations for selected n:
%e A368414   1    6      12     24      30       60        72      120
%e A368414        2*3    2*6    2*12    2*15     2*30      2*36    2*60
%e A368414               3*4    3*8     3*10     3*20      3*24    3*40
%e A368414                      4*6     5*6      4*15      4*18    4*30
%e A368414                              2*3*5    5*12      6*12    5*24
%e A368414                                       6*10      8*9     6*20
%e A368414                                       2*3*10            8*15
%e A368414                                       2*5*6             10*12
%e A368414                                       3*4*5             2*3*20
%e A368414                                                         2*5*12
%e A368414                                                         2*6*10
%e A368414                                                         3*4*10
%e A368414                                                         3*5*8
%e A368414                                                         4*5*6
%t A368414 facs[n_]:=If[n<=1,{{}},Join @@ Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
%t A368414 Table[Length[Select[facs[n], Select[Tuples[First/@FactorInteger[#]&/@#], UnsameQ@@#&]!={}&]],{n,100}]
%Y A368414 For labeled graphs: A133686, complement A367867, A367868, A140638.
%Y A368414 For unlabeled graphs: A134964, complement A140637.
%Y A368414 For set-systems: A367902, ranks A367906, complement A367903, ranks A367907.
%Y A368414 For non-isomorphic set-systems: A368095, complement A368094, A368409.
%Y A368414 Complementary non-isomorphic multiset partitions: A368097, A355529, A368411.
%Y A368414 For non-isomorphic multiset partitions: A368098, A368100.
%Y A368414 The complement is counted by A368413.
%Y A368414 For non-isomorphic set multipartitions: A368422, complement A368421.
%Y A368414 For divisors instead of prime factors: A370813, complement A370814.
%Y A368414 A001055 counts factorizations, strict A045778.
%Y A368414 A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A368414 A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A368414 A283877 counts non-isomorphic set-systems, connected A300913.
%Y A368414 Cf. A340596, A340653, A367769, A367901, A368187, A368412, A368413.
%K A368414 nonn
%O A368414 1,6
%A A368414 _Gus Wiseman_, Dec 29 2023