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A324738
Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.
13
1, 2, 3, 5, 8, 13, 26, 42, 72, 120, 232, 376, 752, 1128, 2256, 4512, 8256, 13632, 27264, 42048, 82944, 158976, 313344, 497664, 995328, 1700352, 3350016, 5815296, 11630592, 17491968, 34983936, 56954880, 108933120, 210788352, 418258944, 804667392, 1609334784
OFFSET
0,2
COMMENTS
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.
LINKS
EXAMPLE
The a(0) = 1 through a(6) = 26 subsets:
{} {} {} {} {} {} {}
{1} {1} {1} {1} {1} {1}
{2} {2} {2} {2} {2}
{3} {3} {3} {3}
{1,3} {4} {4} {4}
{1,3} {5} {5}
{2,4} {1,3} {6}
{3,4} {1,5} {1,3}
{2,4} {1,5}
{2,5} {1,6}
{3,4} {2,4}
{4,5} {2,5}
{2,4,5} {2,6}
{3,4}
{3,6}
{4,5}
{4,6}
{5,6}
{1,3,6}
{1,5,6}
{2,4,5}
{2,4,6}
{2,5,6}
{3,4,6}
{4,5,6}
{2,4,5,6}
MATHEMATICA
Table[Length[Select[Subsets[Range[n]], !MemberQ[#, k_/; SubsetQ[#, PrimePi/@First/@FactorInteger[k]]]&]], {n, 0, 10}]
PROG
(PARI)
pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n, k, if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));
((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
CROSSREFS
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.
Sequence in context: A074030 A336604 A024318 * A132915 A030036 A115212
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 13 2019
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019
STATUS
approved