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

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

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

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

Examples

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

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.

Crossrefs

Cf. A050291-A050296.

Programs

PARI

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

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].

Extensions

More terms from John W. Layman, Oct 25 2002

Corrected and edited by Steven Finch, Feb 25 2009