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A160113
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Number of cubefree integers not exceeding 2^n.
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6
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1, 2, 4, 7, 14, 27, 54, 107, 214, 427, 854, 1706, 3410, 6815, 13629, 27259, 54521, 109042, 218080, 436158, 872318, 1744638, 3489278, 6978546, 13957092, 27914186, 55828364, 111656716, 223313428, 446626866, 893253744, 1786507472, 3573014938
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OFFSET
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0,2
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COMMENTS
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An alternate definition specifying "less than 2^n" would yield the same sequence except for the first 3 terms: 0,1,3,7,14,27,54,107, etc. (since powers of 2 beyond 8 are not cubefree).
The limit of a(n)/2^n is the inverse of Apery's constant, 1/zeta(3) [see A088453].
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LINKS
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G. P. Michon, Table of n, a(n) for n=0..80
G. P. Michon, On the number of cubefree integers not exceeding N.
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FORMULA
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a(n) = Sum for i=1 to 2^(n/3) of A008683(i)*floor(2^n/i^3)
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EXAMPLE
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a(0)=1 because there is just one cubefree integer (1) not exceeding 2^0 = 1.
a(3)=7 because 1,2,3,4,5,6,7 are cubefree but 8 is not.
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MATHEMATICA
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a[n_] := Sum[ MoebiusMu[i]*Floor[2^n/i^3], {i, 1, 2^(n/3)}]; Table[a[n], {n, 0, 32}] (* From Jean-François Alcover, Dec 20 2011, from formula *)
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CROSSREFS
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A004709 (cube-free numbers). A160112 (decimal counterpart for cubefree integers). A143658 (binary counterpart for squarefree integers). A071172 & A053462 (decimal counterpart for squarefree integers).
Sequence in context: A107949 A155099 A136322 * A171231 A094057 A119267
Adjacent sequences: A160110 A160111 A160112 * A160114 A160115 A160116
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Gerard P. Michon, May 02 2009, May 06 2009
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STATUS
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approved
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