Triple-free sets are sets of positive integers defined in weak and strong senses. A set 
is weakly triple-free if 
for every positive
integer 
.
For example, all sets contained in 
are weakly triple-free except 
, 
, 
, and 
(since each of these contains the subset 
). The numbers of weakly triple-free
sets contained in 
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 
that are strongly triple-free are 
, 
, 
, 
, 
, 
, 
, and 
(all other subsets contain
either a double or triple of another set element).
The numbers of strongly triple-free sets contained in 
for 
, 1, 2, ... are 1, 2, 3, 5, 8, 16, 24, 48, 76, 132, ... (OEIS
A050295).
Define
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is a nonnegative integer and 
denotes the cardinal number
of 
.
Then for 
,
2, ..., 
is given by 1, 2, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 12, ... (OEIS A157282),
and 
by 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, ... (OEIS A050296).
Both limits exist (Graham et al. 1977, Finch 2003).
A numerical estimate for the weakly triple-free set constant is
 |
(3)
|
This estimate is reported in OEIS A386439. The earlier value 0.8003194838 was a conditional estimate based on a conjecture of Chung et al. (2002). The strongly triple-free set constant is
 |
(4)
|
(OEIS A086316; Cassaigne and Zimmermann 1996, Finch 2003).
-fold Free Sets of Positive Integers." Math. Mag. 68,
71-72, 1995.Sloane, N. J. A. Sequences