

A050294


Maximum cardinality of a 3foldfree subset of {1, 2, ..., n}.


2



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

1,2


COMMENTS

For a given r>1, a set is rfoldfree if it does not contain any subset of the form {x, r*x}.
If r is in A050376, then an rfoldfree 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 rfoldfree 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.
Equals A051068 shifted by 1.  Michel Dekking, Feb 18 2019


LINKS

Table of n, a(n) for n=1..73.
Steven R. Finch, TripleFree Sets of Integers [From Steven Finch, Apr 20 2019]
Bruce Reznick, Problem 1440, Mathematics Magazine, Vol. 67 (1994).
B. Reznick and R. Holzsager, rfold free sets of positive integers, Math. Magazine 68 (1995) 7172.
Eric Weisstein's World of Mathematics, TripleFree Set.


FORMULA

Take r = 3 in a(n) = (r n + sum [k = 0 to m] (1)^k b(k)) / (r + 1), where [b(m) b(m1) ... b(0)] is the baser representation of n.  Rob Pratt, Apr 21 2004
Take r=3 in a(n) = na(floor(n/r)); a(n)=nfloor(n/r)+floor(n/r^2)floor(n/r^3)+... [Vladimir Shevelev, Feb 22 2011].


EXAMPLE

a(26)=26a(floor(26/3))=26a(8)=266=20.


PROG

(PARI) a(n)=if(n==0, 0, na(floor(n/3))); \\ Joerg Arndt, Apr 27 2013


CROSSREFS

Cf. A050291A050296.
Sequence in context: A175268 A241948 A051068 * A097950 A011885 A211524
Adjacent sequences: A050291 A050292 A050293 * A050295 A050296 A050297


KEYWORD

nonn


AUTHOR

Eric W. Weisstein


EXTENSIONS

More terms from John W. Layman, Oct 25 2002
Corrected and edited by Steven Finch, Feb 25 2009


STATUS

approved



