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A370585
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Number of maximal subsets of {1..n} such that it is possible to choose a different prime factor of each element.
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22
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1, 1, 1, 1, 2, 2, 5, 5, 7, 11, 25, 25, 38, 38, 84, 150, 178, 178, 235, 235, 341, 579, 1235, 1235
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OFFSET
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0,5
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COMMENTS
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First differs from A307984 at a(21) = 579, A307984(21) = 578. The difference is due to the set {10,11,13,14,15,17,19,21}, which is not a basis because log(10) + log(21) = log(14) + log(15).
Also length-pi(n) subsets of {1..n} such that it is possible to choose a different prime factor of each element.
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LINKS
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EXAMPLE
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The a(0) = 1 through a(8) = 7 subsets:
{} {} {2} {2,3} {2,3} {2,3,5} {2,3,5} {2,3,5,7} {2,3,5,7}
{3,4} {3,4,5} {2,5,6} {2,5,6,7} {2,5,6,7}
{3,4,5} {3,4,5,7} {3,4,5,7}
{3,5,6} {3,5,6,7} {3,5,6,7}
{4,5,6} {4,5,6,7} {3,5,7,8}
{4,5,6,7}
{5,6,7,8}
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MATHEMATICA
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Table[Length[Select[Subsets[Range[n], {PrimePi[n]}], Length[Select[Tuples[If[#==1, {}, First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]], {n, 0, 10}]
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CROSSREFS
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Factorizations of this type are counted by A368414, complement A368413.
A307984 counts Q-bases of logarithms of positive integers.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A005117, A045778, A133686, A333331, A355739, A355740, A355744, A355745, A367905, A368110.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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