A set of positive integers is called weakly triple-free if, for any integer , the set {x,2x,3x} !subset= S" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline2.gif" style="height:15px; width:90px" />. For example, all subsets of {1,2,3,4,5}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline3.gif" style="height:15px; width:77px" /> are weakly triple-free except {1,2,3}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline4.gif" style="height:15px; width:47px" />, {1,2,3,4}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline5.gif" style="height:15px; width:62px" />, {1,2,3,5}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline6.gif" style="height:15px; width:62px" />, and {1,2,3,4,5}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline7.gif" style="height:15px; width:77px" /> (since each of these contains the subset {1,2,3}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline8.gif" style="height:15px; width:47px" /> The numbers of weakly triple-free subsets of {1,2,...,n}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline9.gif" style="height:15px; width:69px" /> for , 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  implies  and . For example, the only subsets of {1,2,3,4}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline14.gif" style="height:15px; width:62px" /> that are strongly triple-free are , {1}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline16.gif" style="height:15px; width:17px" />, {2}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline17.gif" style="height:15px; width:17px" />, {3}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline18.gif" style="height:15px; width:17px" />, {4}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline19.gif" style="height:15px; width:17px" />, {1,4}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline20.gif" style="height:15px; width:32px" />, {2,3}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline21.gif" style="height:15px; width:32px" />, and {3,4}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline22.gif" style="height:15px; width:32px" /> (all other subsets contain either a double or triple of another set element). The numbers of strongly triple-free subsets for , 1, 2, ... are 1, 2, 3, 5, 8, 16, 24, 48, 76, 132, ... (OEIS A050295).

Define

		{|S|:S subset {1,2,...,n} is weakly triple-free}" src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline26.gif" style="height:15px; width:278px" />	(1)
		{|S|:S subset {1,2,...,n} is strongly triple-free}," src="https://mathworld.wolfram.com/images/equations/Triple-FreeSet/Inline29.gif" style="height:15px; width:288px" />	(2)

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

(3)

(conjectured) and

(4)

(OEIS A086316; Finch 2003).

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. https://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. "-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."