TOPICS
Search

Triple-Free Set


A set of positive integers is called weakly triple-free if, for any integer x, the set {x,2x,3x} !subset= S. For example, all subsets of {1,2,3,4,5} are weakly triple-free except {1,2,3}, {1,2,3,4}, {1,2,3,5}, and {1,2,3,4,5} (since each of these contains the subset {1,2,3} The numbers of weakly triple-free subsets of {1,2,...,n} for n=0, 1,2, ... are 1, 2, 4, 7, 14, 28, 50, 100, 200, 360, 720, ... (OEIS A068060).

A set of positive integers is called strongly triple-free if x in S implies 2x not in S and 3x not in S. For example, the only subsets of {1,2,3,4} that are strongly triple-free are emptyset, {1}, {2}, {3}, {4}, {1,4}, {2,3}, and {3,4} (all other subsets contain either a double or triple of another set element). The numbers of strongly triple-free subsets for n=0, 1, 2, ... are 1, 2, 3, 5, 8, 16, 24, 48, 76, 132, ... (OEIS A050295).

Triple-FreeSet

Define

p(n)=max{|S|:S subset {1,2,...,n} is weakly triple-free}
(1)
q(n)=max{|S|:S subset {1,2,...,n} is strongly triple-free},
(2)

where |S| denotes the cardinal number of (number of members in) S. Then for n=1, 2, ..., p(n) is given by 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 11, ... (OEIS A157282), and q(n) by 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, ... (OEIS A050296). Asymptotic formulas are given by

 lim_(n->infty)(p(n))/n=0.8003194838
(3)

(conjectured) and

 lim_(n->infty)(q(n))/n=0.6134752692...
(4)

(OEIS A086316; Finch 2003).


See also

A-Sequence, Double-Free Set, Sum-Free Set

Explore with Wolfram|Alpha

References

Chung, F.; Erdős, P.; and Graham, R. "On Sparse Sets Hitting Linear Forms." In Number Theory for the Millennium, vol. 1, Proc. 2000 Urbana Conf. (Ed. M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand, and W. Philipp). Natick, MA: A K Peters, pp. 257-272, 2002.Finch,  S. "Triple-Free Sets of Integers." Sep. 5, 2002. http://algo.inria.fr/csolve/triple/.Finch, S. R. "Triple-Free Set Constants." §2.26 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 183-185, 2003.Graham, R.; Spencer, J.; and Witsenhausen, H. "On Extremal Density Theorems for Linear Forms." In Number Theory and Algebra (Ed. H. Zassenhaus). New York: Academic Press, pp. 103-109, 1977.Reznick, B. and Holzsager, R. "r-fold Free Sets of Positive Integers." Math. Mag. 68, 71-72, 1995.Sloane, N. J. A. Sequences A050295, A050296, A068060, A086316, and A157282 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Triple-Free Set

Cite this as:

Weisstein, Eric W. "Triple-Free Set." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Triple-FreeSet.html

Subject classifications