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A324739 Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.

%I A324739 #8 Aug 16 2019 14:48:26
%S A324739 1,2,3,6,10,20,30,60,96,192,312,624,936,1872,3744,7488,12480,24960,
%T A324739 37440,74880,142848,285696,456192,912384,1548288,3096576,5308416,
%U A324739 10616832,15925248,31850496,51978240,103956480,200835072,401670144,771489792,1542979584,2314469376
%N A324739 Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.
%C A324739 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 A324739 Andrew Howroyd, <a href="/A324739/b324739.txt">Table of n, a(n) for n = 1..100</a>
%e A324739 The a(1) = 1 through a(6) = 20 subsets:
%e A324739   {}  {}   {}   {}     {}       {}
%e A324739       {2}  {2}  {2}    {2}      {2}
%e A324739            {3}  {3}    {3}      {3}
%e A324739                 {4}    {4}      {4}
%e A324739                 {2,4}  {5}      {5}
%e A324739                 {3,4}  {2,4}    {6}
%e A324739                        {2,5}    {2,4}
%e A324739                        {3,4}    {2,5}
%e A324739                        {4,5}    {2,6}
%e A324739                        {2,4,5}  {3,4}
%e A324739                                 {3,6}
%e A324739                                 {4,5}
%e A324739                                 {4,6}
%e A324739                                 {5,6}
%e A324739                                 {2,4,5}
%e A324739                                 {2,4,6}
%e A324739                                 {2,5,6}
%e A324739                                 {3,4,6}
%e A324739                                 {4,5,6}
%e A324739                                 {2,4,5,6}
%t A324739 Table[Length[Select[Subsets[Range[2,n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,10}]
%o A324739 (PARI)
%o A324739 pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
%o A324739 a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
%o A324739 ((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 A324739 The maximal case is A324762. The case of subsets of {1...n} is A324738. The strict integer partition version is A324750. The integer partition version is A324755. The Heinz number version is A324760. An infinite version is A324694.
%Y A324739 Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
%Y A324739 Cf. A324695, A324696, A324736, A324737, A324741, A324742, A324744, A324764.
%K A324739 nonn
%O A324739 1,2
%A A324739 _Gus Wiseman_, Mar 14 2019
%E A324739 Terms a(21) and beyond from _Andrew Howroyd_, Aug 16 2019