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Double-Free Set

A Set of Positive integers is double-free if, for any integer $x$, the Set $\{x, 2x\}\not\subset S$ (or equivalently, $x\in S$ Implies $2x\not\in S$). Define

r(n)=\max\{S: S\subset \{1, 2, \ldots, n\}\hbox{ is double-free}\}.

Then an asymptotic formula is

r(n)\sim {\textstyle{2\over 3}}n+{\mathcal O}(\ln n)

(Wang 1989).

See also Triple-Free Set


Finch, S. ``Favorite Mathematical Constants.''

Wang, E. T. H. ``On Double-Free Sets of Integers.'' Ars Combin. 28, 97-100, 1989.

© 1996-9 Eric W. Weisstein