A set of positive integers is called weakly triple-free if, for any integer 
, the set 
. For example, all subsets of 
are weakly triple-free except 
, 
, 
, and 
(since each of these contains the subset 
The numbers of weakly triple-free subsets of 
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 subsets for 
, 1, 2, ... are 1, 2, 3, 5, 8, 16, 24, 48, 76, 132, ... (OEIS
A050295).
Define
 |  |  |
(1)
|
 |  |  |
(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).
-fold Free Sets of Positive Integers." Math. Mag. 68,
71-72, 1995.Sloane, N. J. A. Sequences