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%I #9 Aug 16 2019 14:48:51
%S 1,2,3,5,8,13,19,30,54,96,156,248,440,688,1120,1864,3664,5856,11232,
%T 16896,31296,53952,91008,137472,270528,516720,863088,1710816,3173856,
%U 4836672,9329472,14897376,29788128,52256448,88429248,166037184,331648704,497685888,829449600
%N Number of subsets of {1...n} containing no prime indices of the elements.
%C 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 Andrew Howroyd, <a href="/A324741/b324741.txt">Table of n, a(n) for n = 0..100</a>
%e The a(0) = 1 through a(6) = 19 subsets:
%e {} {} {} {} {} {} {}
%e {1} {1} {1} {1} {1} {1}
%e {2} {2} {2} {2} {2}
%e {3} {3} {3} {3}
%e {1,3} {4} {4} {4}
%e {1,3} {5} {5}
%e {2,4} {1,3} {6}
%e {3,4} {1,5} {1,3}
%e {2,4} {1,5}
%e {2,5} {2,4}
%e {3,4} {2,5}
%e {4,5} {3,4}
%e {2,4,5} {3,6}
%e {4,5}
%e {4,6}
%e {5,6}
%e {2,4,5}
%e {3,4,6}
%e {4,5,6}
%e An example for n = 20 is {5,6,7,9,10,12,14,15,16,19,20}, with prime indices:
%e 5: {3}
%e 6: {1,2}
%e 7: {4}
%e 9: {2,2}
%e 10: {1,3}
%e 12: {1,1,2}
%e 14: {1,4}
%e 15: {2,3}
%e 16: {1,1,1,1}
%e 19: {8}
%e 20: {1,1,3}
%e None of these prime indices {1,2,3,4,8} belong to the subset, as required.
%t Table[Length[Select[Subsets[Range[n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,10}]
%o (PARI)
%o pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
%o a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
%o ((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitand(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<<k)), t)); t))(1,0)} \\ _Andrew Howroyd_, Aug 16 2019
%Y The maximal case is A324743. The strict integer partition version is A324751. The integer partition version is A324756. The Heinz number version is A324758. An infinite version is A304360.
%Y Cf. A000720, A001462, A007097, A076078, A084422, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
%Y Cf. A324695, A324736, A324742.
%K nonn
%O 0,2
%A _Gus Wiseman_, Mar 15 2019
%E Terms a(21) and beyond from _Andrew Howroyd_, Aug 16 2019
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