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A324738 Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.

%I A324738 #8 Aug 16 2019 14:47:53
%S A324738 1,2,3,5,8,13,26,42,72,120,232,376,752,1128,2256,4512,8256,13632,
%T A324738 27264,42048,82944,158976,313344,497664,995328,1700352,3350016,
%U A324738 5815296,11630592,17491968,34983936,56954880,108933120,210788352,418258944,804667392,1609334784
%N A324738 Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.
%C A324738 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.
%H A324738 Andrew Howroyd, <a href="/A324738/b324738.txt">Table of n, a(n) for n = 0..100</a>
%e A324738 The a(0) = 1 through a(6) = 26 subsets:
%e A324738   {}  {}   {}   {}     {}     {}       {}
%e A324738       {1}  {1}  {1}    {1}    {1}      {1}
%e A324738            {2}  {2}    {2}    {2}      {2}
%e A324738                 {3}    {3}    {3}      {3}
%e A324738                 {1,3}  {4}    {4}      {4}
%e A324738                        {1,3}  {5}      {5}
%e A324738                        {2,4}  {1,3}    {6}
%e A324738                        {3,4}  {1,5}    {1,3}
%e A324738                               {2,4}    {1,5}
%e A324738                               {2,5}    {1,6}
%e A324738                               {3,4}    {2,4}
%e A324738                               {4,5}    {2,5}
%e A324738                               {2,4,5}  {2,6}
%e A324738                                        {3,4}
%e A324738                                        {3,6}
%e A324738                                        {4,5}
%e A324738                                        {4,6}
%e A324738                                        {5,6}
%e A324738                                        {1,3,6}
%e A324738                                        {1,5,6}
%e A324738                                        {2,4,5}
%e A324738                                        {2,4,6}
%e A324738                                        {2,5,6}
%e A324738                                        {3,4,6}
%e A324738                                        {4,5,6}
%e A324738                                        {2,4,5,6}
%t A324738 Table[Length[Select[Subsets[Range[n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,10}]
%o A324738 (PARI)
%o A324738 pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
%o A324738 a(n)={my(p=vector(n,k,if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));
%o A324738 ((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<<k)), t)); t))(1,0)} \\ _Andrew Howroyd_, Aug 16 2019
%Y A324738 The maximal case is A324744. The case of subsets of {2...n} is A324739. The strict integer partition version is A324749. The integer partition version is A324754. The Heinz number version is A324759. An infinite version is A324694.
%Y A324738 Cf. A000720, A001221, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
%Y A324738 Cf. A324695, A324736, A324741, A324750, A324755, A324760.
%K A324738 nonn
%O A324738 0,2
%A A324738 _Gus Wiseman_, Mar 13 2019
%E A324738 Terms a(21) and beyond from _Andrew Howroyd_, Aug 16 2019