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A324763
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Number of maximal subsets of {2...n} containing no prime indices of the elements.
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10
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1, 1, 2, 2, 2, 3, 6, 6, 6, 6, 10, 10, 16, 16, 16, 16, 24, 24, 48, 48, 48, 48, 84, 84, 84, 84, 84, 84, 144, 144, 228, 228, 228, 228, 228, 228, 420, 420, 420, 420, 648, 648, 1080, 1080, 1080, 1080, 1800, 1800, 1800, 1800, 1800, 1800, 3600, 3600, 3600, 3600, 3600
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(9) = 6 subsets:
{} {2} {2} {2,4} {3,4} {2,4,5} {2,4,5} {2,4,5,8} {2,4,5,8}
{3} {3,4} {2,4,5} {3,4,6} {2,5,7} {2,5,7,8} {2,5,7,8}
{4,5,6} {3,4,6} {3,4,6,8} {3,4,6,8,9}
{3,6,7} {3,6,7,8} {3,6,7,8,9}
{4,5,6} {4,5,6,8} {4,5,6,8,9}
{5,6,7} {5,6,7,8} {5,6,7,8,9}
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MATHEMATICA
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maxim[s_]:=Complement[s, Last/@Select[Tuples[s, 2], UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[2, n]], Intersection[#, PrimePi/@First/@Join@@FactorInteger/@#]=={}&]]], {n, 10}]
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PROG
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(PARI)
pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1, k, pset(k+1)>>1), 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
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CROSSREFS
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The non-maximal version is A324742.
The version for subsets of {1...n} is A324741.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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