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

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/triple/triple.html

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