A set of positive integers is double-free if, for any integer , the set {x,2x} !subset= S" src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/Inline2.gif" style="height:15px; width:65px" /> (or equivalently,  implies ). For example, of the subsets of {1,2,3}" src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/Inline5.gif" style="height:15px; width:47px" />, the sets , {1}" src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/Inline7.gif" style="height:15px; width:17px" />, {2}" src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/Inline8.gif" style="height:15px; width:17px" />, {2,3}" src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/Inline9.gif" style="height:15px; width:32px" />, {1,3}" src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/Inline10.gif" style="height:15px; width:32px" />, and {3}" src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/Inline11.gif" style="height:15px; width:17px" /> are double-free, while {1,2}" src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/Inline12.gif" style="height:15px; width:32px" /> and {1,2,3}" src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/Inline13.gif" style="height:15px; width:47px" /> are not.

The number  of double-free subsets of {1,2,...,n}" src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/Inline15.gif" style="height:15px; width:69px" /> can be computed using  and the recurrence relation

(1)

where  is a Fibonacci number, 1, 1, 2, 3, 5, 8, ... (OEIS A000045), and  is the binary carry sequence giving the number of trailing 0s in the binary representation of . For , 2, ...,  is given by 0, 1, 0, 2, 0, 1, 3, 0, 1, ... (OEIS A007814), while the corresponding  are 2, 3, 6, 10, 20, 30, 60, 96, 192, ... (OEIS A050291).

Define

{|S|:S subset {1,2,...,n} is double-free}, " src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/NumberedEquation2.gif" style="height:15px; width:288px" />

(2)

where  is the cardinal number of (number of members in) . Then for , 2, ...,  is given by 1, 1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, ... (OEIS A050292). An explicit formula for  is given by

(3)

where

{1 if b(i) is even; 0 if b(i) is odd, " src="https://mathworld.wolfram.com/images/equations/Double-FreeSet/NumberedEquation4.gif" style="height:41px; width:152px" />

(4)

if the characteristic function of  (defined above), and the first few values of  are 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, ... (OEIS A035263). A simple recurrence relation for  is given by

(5)

with  (Wang 1989), where  is the floor function and  is the ceiling function. An asymptotic formula for  is given by

(6)

(Wang 1989).

REFERENCES:

Finch, S. R. "Triple-Free Set Constants." §2.26 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 183-185, 2003.

Sloane, N. J. A. Sequences A000045/M0692, A007814, A035263, A050291 and A050292 in "The On-Line Encyclopedia of Integer Sequences."

Wang, E. T. H. "On Double-Free Sets of Integers." Ars Combin. 28, 97-100, 1989.