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A370584 Number of subsets of {1..n} such that only one set can be obtained by choosing a different prime factor of each element.

%I A370584 #11 Mar 28 2025 13:42:31
%S A370584 1,1,2,4,6,12,18,36,48,68,104,208,284,568,888,1296,1548,3096,3968,
%T A370584 7936,10736,15440,24008,48016,58848,73680,114368,132608,176240,352480,
%U A370584 449824,899648,994976,1399968,2160720,2859584,3296048,6592096,10156672,14214576,16892352
%N A370584 Number of subsets of {1..n} such that only one set can be obtained by choosing a different prime factor of each element.
%C A370584 For example, the only choice of a different prime factor of each element of (4,5,6) is (2,5,3).
%e A370584 The a(0) = 1 through a(6) = 18 subsets:
%e A370584   {}  {}  {}   {}     {}     {}       {}
%e A370584           {2}  {2}    {2}    {2}      {2}
%e A370584                {3}    {3}    {3}      {3}
%e A370584                {2,3}  {4}    {4}      {4}
%e A370584                       {2,3}  {5}      {5}
%e A370584                       {3,4}  {2,3}    {2,3}
%e A370584                              {2,5}    {2,5}
%e A370584                              {3,4}    {2,6}
%e A370584                              {3,5}    {3,4}
%e A370584                              {4,5}    {3,5}
%e A370584                              {2,3,5}  {3,6}
%e A370584                              {3,4,5}  {4,5}
%e A370584                                       {4,6}
%e A370584                                       {2,3,5}
%e A370584                                       {2,5,6}
%e A370584                                       {3,4,5}
%e A370584                                       {3,5,6}
%e A370584                                       {4,5,6}
%t A370584 Table[Length[Select[Subsets[Range[n]], Length[Union[Sort/@Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]],{n,0,10}]
%Y A370584 For divisors instead of factors we have A051026, cf. A368110, A355740.
%Y A370584 The version for set-systems is A367904, ranks A367908.
%Y A370584 Multisets of this type are ranked by A368101, cf. A368100, A355529.
%Y A370584 For existence we have A370582, differences A370586.
%Y A370584 For nonexistence we have A370583, differences A370587.
%Y A370584 Maximal sets of this type are counted by A370585.
%Y A370584 The version for partitions is A370594, cf. A370592, A370593.
%Y A370584 For binary indices instead of factors we have A370638, cf. A370636, A370637.
%Y A370584 The version for factorizations is A370645, cf. A368414, A368413.
%Y A370584 For unlabeled multiset partitions we have A370646, cf. A368098, A368097.
%Y A370584 A006530 gives greatest prime factor, least A020639.
%Y A370584 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370584 A355741 counts ways to choose a prime factor of each prime index.
%Y A370584 Cf. A000040, A000720, A003963, A005117, A045778, A133686, A307984, A355739, A355744, A355745, A367905.
%K A370584 nonn
%O A370584 0,3
%A A370584 _Gus Wiseman_, Feb 26 2024
%E A370584 More terms from _Jinyuan Wang_, Mar 28 2025