cp's OEIS Frontend

A370585 Number of maximal subsets of {1..n} such that it is possible to choose a different prime factor of each element.

%I A370585 #10 Feb 14 2025 05:06:28
%S A370585 1,1,1,1,2,2,5,5,7,11,25,25,38,38,84,150,178,178,235,235,341,579,1235,
%T A370585 1235,1523,1968,4160,4824,6840,6840,9140,9140,10028,16264,33956,48680,
%U A370585 56000,56000,116472,186724,223884,223884,290312,290312,403484,484028,1001420
%N A370585 Number of maximal subsets of {1..n} such that it is possible to choose a different prime factor of each element.
%C A370585 First differs from A307984 at a(21) = 579, A307984(21) = 578. The difference is due to the set {10,11,13,14,15,17,19,21}, which is not a basis because log(10) + log(21) = log(14) + log(15).
%C A370585 Also length-pi(n) subsets of {1..n} such that it is possible to choose a different prime factor of each element.
%e A370585 The a(0) = 1 through a(8) = 7 subsets:
%e A370585   {}  {}  {2}  {2,3}  {2,3}  {2,3,5}  {2,3,5}  {2,3,5,7}  {2,3,5,7}
%e A370585                       {3,4}  {3,4,5}  {2,5,6}  {2,5,6,7}  {2,5,6,7}
%e A370585                                       {3,4,5}  {3,4,5,7}  {3,4,5,7}
%e A370585                                       {3,5,6}  {3,5,6,7}  {3,5,6,7}
%e A370585                                       {4,5,6}  {4,5,6,7}  {3,5,7,8}
%e A370585                                                           {4,5,6,7}
%e A370585                                                           {5,6,7,8}
%t A370585 Table[Length[Select[Subsets[Range[n], {PrimePi[n]}],Length[Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]],{n,0,10}]
%Y A370585 Multisets of this type are ranked by A368100, complement A355529.
%Y A370585 Factorizations of this type are counted by A368414, complement A368413.
%Y A370585 The version for set-systems is A368601, max of A367902 (complement A367903).
%Y A370585 This is the maximal case of A370582, complement A370583, cf. A370584.
%Y A370585 A different kind of maximality is A370586, complement A370587.
%Y A370585 The case containing n is A370590, complement A370591.
%Y A370585 Partitions of this type (choosable) are A370592, complement A370593.
%Y A370585 For binary indices instead of factors we have A370640, cf. A370636, A370637.
%Y A370585 A006530 gives greatest prime factor, least A020639.
%Y A370585 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370585 A307984 counts Q-bases of logarithms of positive integers.
%Y A370585 A355741 counts choices of a prime factor of each prime index.
%Y A370585 Cf. A000040, A000720, A005117, A045778, A133686, A333331, A355739, A355740, A355744, A355745, A367905, A368110.
%K A370585 nonn
%O A370585 0,5
%A A370585 _Gus Wiseman_, Feb 26 2024
%E A370585 More terms from _Jinyuan Wang_, Feb 14 2025