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A324743
Number of maximal subsets of {1...n} containing no prime indices of the elements.
17
1, 1, 2, 2, 3, 4, 5, 8, 8, 8, 8, 12, 12, 18, 18, 19, 19, 30, 30, 54, 54, 54, 54, 96, 96, 96, 96, 96, 96, 156, 156, 244, 244, 248, 248, 248, 248, 440, 440, 440, 440, 688, 688, 1120, 1120, 1120, 1120, 1864, 1864, 1864, 1864, 1864, 1864, 3664, 3664, 3664, 3664, 3664
OFFSET
0,3
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(8) = 8 maximal subsets:
{} {1} {1} {2} {1,3} {1,3} {1,3} {1,3,7} {1,3,7}
{2} {1,3} {2,4} {1,5} {1,5} {1,5,7} {1,5,7}
{3,4} {3,4} {2,4,5} {2,4,5} {2,4,5,8}
{2,4,5} {3,4,6} {2,5,7} {2,5,7,8}
{4,5,6} {3,4,6} {3,4,6,8}
{3,6,7} {3,6,7,8}
{4,5,6} {4,5,6,8}
{5,6,7} {5,6,7,8}
An example for n = 15 is {1,5,7,9,13,15}, with prime indices:
1: {}
5: {3}
7: {4}
9: {2,2}
13: {6}
15: {2,3}
None of these prime indices {2,3,4,6} belong to the subset, as required.
MATHEMATICA
maxim[s_]:=Complement[s, Last/@Select[Tuples[s, 2], UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]], Intersection[#, PrimePi/@First/@Join@@FactorInteger/@#]=={}&]]], {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, pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=my(e=0); forstep(k=#p, 1, -1, if(bittest(b, k), e=bitor(e, p[k]), if(!bittest(e, k) && !bitand(p[k], b), return(0)) )); 1);
((k, b)->if(k>#p, ismax(b), my(f=!bitand(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<<k)))))(1, 0)} \\ Andrew Howroyd, Aug 26 2019
CROSSREFS
The non-maximal case is A324741. The case for subsets of {2...n} is A324763.
Sequence in context: A175306 A359348 A021993 * A240011 A211228 A338463
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 15 2019
EXTENSIONS
Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019
STATUS
approved