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A050294 Maximum cardinality of a 3-fold-free subset of {1, 2, ..., n}. 2

%I

%S 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,

%T 21,22,22,23,24,24,25,26,27,28,29,29,30,31,31,32,33,34,35,36,36,37,38,

%U 38,39,40,40,41,42,42,43,44,44,45,46,47,48,49,49,50,51,51,52,53,54,55

%N Maximum cardinality of a 3-fold-free subset of {1, 2, ..., n}.

%C For a given r>1, a set is r-fold-free if it does not contain any subset of the form {x, r*x}.

%C 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.

%C Equals A051068 shifted by 1. - _Michel Dekking_, Feb 18 2019

%H Steven R. Finch, <a href="/FinchTriple.html">Triple-Free Sets of Integers</a> [From Steven Finch, Apr 20 2019]

%H Bruce Reznick, <a href="http://www.jstor.org/stable/2690562">Problem 1440</a>, Mathematics Magazine, Vol. 67 (1994).

%H B. Reznick and R. Holzsager, <a href="http://www.jstor.org/stable/2691384">r-fold free sets of positive integers</a>, Math. Magazine 68 (1995) 71-72.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Triple-FreeSet.html">Triple-Free Set.</a>

%F 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_, Apr 21 2004

%F 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].

%e a(26)=26-a(floor(26/3))=26-a(8)=26-6=20.

%o (PARI) a(n)=if(n==0,0,n-a(floor(n/3))); \\ _Joerg Arndt_, Apr 27 2013

%Y Cf. A050291-A050296.

%K nonn

%O 1,2

%A _Eric W. Weisstein_

%E More terms from _John W. Layman_, Oct 25 2002

%E Corrected and edited by _Steven Finch_, Feb 25 2009

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Last modified July 29 08:46 EDT 2021. Contains 346340 sequences. (Running on oeis4.)