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Triple-Free Set

A Set of Positive integers is called weakly triple-free if, for any integer $x$, the Set $\{x, 2x,
3x\}\not\subset S$. It is called strongly triple-free if $x\in S$ Implies $2x\not\in S$ and $3x\not\in S$. Define

p(n)&=&\max\{\vert S\vert: S\subset \{1, 2, \ldots, n\} \hbox{...
...set \{1, 2, \ldots, n\} \hbox{\rm\ is\ strongly\ triple-free}\},

where $\vert S\vert$ denotes the Cardinality of $S$, then

\lim_{n\to\infty} {p(n)\over n}\geq {\textstyle{4\over 5}}


\lim_{n\to\infty} {q(n)\over n}=0.6134752692\ldots


See also Double-Free Set


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

© 1996-9 Eric W. Weisstein