OFFSET
1,2
COMMENTS
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.
Equals A051068 shifted by 1. - Michel Dekking, Feb 18 2019
LINKS
Steven R. Finch, Triple-Free Sets of Integers [From Steven Finch, Apr 20 2019]
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.
Eric Weisstein's World of Mathematics, Triple-Free 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(m-1) ... b(0)] is the base-r representation of n. - Rob Pratt, 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].
EXAMPLE
a(26)=26-a(floor(26/3))=26-a(8)=26-6=20.
PROG
(PARI) a(n)=if(n==0, 0, n-a(floor(n/3))); \\ Joerg Arndt, Apr 27 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from John W. Layman, Oct 25 2002
Corrected and edited by Steven Finch, Feb 25 2009
STATUS
approved