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A324737 Number of subsets of {2...n} containing every element of {2...n} whose prime indices all belong to the subset.

%I A324737 #10 Aug 24 2019 21:24:24
%S A324737 1,2,3,6,8,16,24,48,84,168,216,432,648,1296,2448,4896,6528,13056,
%T A324737 19584,39168,77760,155520,229248,458496,790272,1580544,3128832,
%U A324737 6257664,9386496,18772992,24081408,48162816,95938560,191877120,378335232,756670464,1135005696,2270011392
%N A324737 Number of subsets of {2...n} containing every element of {2...n} whose prime indices all belong to the subset.
%C A324737 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A324737 Also the number of subsets of {2...n} with complement containing no term whose prime indices all belong to the subset.
%H A324737 Andrew Howroyd, <a href="/A324737/b324737.txt">Table of n, a(n) for n = 1..100</a>
%e A324737 The a(1) = 1 through a(6) = 16 subsets:
%e A324737   {}  {}   {}     {}       {}         {}
%e A324737       {2}  {3}    {3}      {4}        {4}
%e A324737            {2,3}  {4}      {5}        {5}
%e A324737                   {2,3}    {3,5}      {6}
%e A324737                   {3,4}    {4,5}      {3,5}
%e A324737                   {2,3,4}  {2,3,5}    {4,5}
%e A324737                            {3,4,5}    {4,6}
%e A324737                            {2,3,4,5}  {5,6}
%e A324737                                       {2,3,5}
%e A324737                                       {3,4,5}
%e A324737                                       {3,5,6}
%e A324737                                       {4,5,6}
%e A324737                                       {2,3,4,5}
%e A324737                                       {2,3,5,6}
%e A324737                                       {3,4,5,6}
%e A324737                                       {2,3,4,5,6}
%e A324737 An example for n = 15 is {2, 3, 5, 8, 9, 10, 11, 15}. The numbers from 2 to 15 with all prime indices in the subset are {3, 5, 9, 11, 15}, which all belong to the subset, as required.
%t A324737 Table[Length[Select[Subsets[Range[2,n]],Function[set,SubsetQ[set,Select[Range[2,n],SubsetQ[set,PrimePi/@First/@FactorInteger[#]]&]]]]],{n,10}]
%o A324737 (PARI)
%o A324737 pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
%o A324737 a(n)={my(p=vector(n-1, k, pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
%o A324737 ((k, b)->if(k>#p, 1, my(t=self()(k+1, b+(1<<k))); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b), t)); t))(1, 0)} \\ _Andrew Howroyd_, Aug 24 2019
%Y A324737 Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A290689, A290822, A304360, A306844.
%Y A324737 Cf. A324697, A324698, A324736, A324738, A324748, A324753, A324755.
%K A324737 nonn
%O A324737 1,2
%A A324737 _Gus Wiseman_, Mar 13 2019
%E A324737 Terms a(21) and beyond from _Andrew Howroyd_, Aug 24 2019