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A050294
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Maximum cardinality of a 3-fold-free subset of {1, 2, ..., n}.
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2
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1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 11, 12, 13, 14, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 27, 28, 29, 29, 30, 31, 31, 32, 33, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 47, 48, 49, 49, 50, 51, 51, 52, 53, 54, 55
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OFFSET
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1,2
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COMMENTS
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For a given r>1, a set is r-fold-free if it does not contain any subset of the form {x, r*x}.
If r is in A050376, then an r-fold-free set with the highest cardinality is obtained by removing from {1,...,n} all numbers for which r is an infinitary divisor (for the definition of the infinitary divisor of n, see comment to A037445). In general, an r-fold-free set with the highest cardinality is obtained by removing from {1,...,n} all numbers for which r is an oex divisor (for the definition of the oex divisor of n, see A186643). - Vladimir Shevelev Feb 22 2011 and Feb 28 2011.
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REFERENCES
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Bruce Reznick, Problem 1440, Mathematics Magazine, Vol. 67 (1994).
B. Reznick and R. Holzsager, r-fold free sets of positive integers, Math. Magazine 68 (1995) 71-72.
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LINKS
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Table of n, a(n) for n=1..73.
S. R. Finch, Triple-Free Sets of Integers
Eric Weisstein's World of Mathematics, Triple-Free Set.
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FORMULA
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Take r = 3 in a(n) = (r n + sum [k = 0 to m] (-1)^k b(k)) / (r + 1), where [b(m) b(m-1) ... b(0)] is the base-r representation of n. - Rob Pratt (Rob.Pratt(AT)sas.com), Apr 21 2004
Take r=3 in a(n)=n-a(floor(n/r)); a(n)=n-floor(n/r)+floor(n/r^2)-floor(n/r^3)+... [Vladimir Shevelev, Feb 22 2011].
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EXAMPLE
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a(26)=26-a(floor(26/3))=26-a(8)=26-6=20.
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PROG
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(PARI) a(n)=if(n==0, 0, n-a(floor(n/3))); \\ Joerg Arndt, Apr 27 2013
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CROSSREFS
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Cf. A050291-A050296.
Sequence in context: A127038 A175268 A051068 * A097950 A011885 A211524
Adjacent sequences: A050291 A050292 A050293 * A050295 A050296 A050297
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein
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EXTENSIONS
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More terms from John W. Layman, Oct 25 2002
Corrected and edited by Steven Finch, Feb 25 2009
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STATUS
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approved
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