A nondividing set is a set of positive integers in which no element divides the sum of any nonempty
subset of the other elements. For example, is dividing, since (and ), but is nondividing since 4 divides none of , and similarly for 6 and 7. The empty
set and sets of length one are therefore trivially nondividing.
Also, any set other than which contains 1 is dividing.
Consider all possible subsets of the integers in .
Then the numbers of nondividing subsets on , , ... are 1, 2, 3, 5, 7, 11, 14, 21, ... (OEIS A051014).
For example, the 14 nondividing sets in are , , , , , , , , , , , , , and .
Abbott, H. L. "Extremal Problems on Non-Averaging and Non-Dividing Sets." Pacific J. Math.91, 1-12, 1980.Guy,
R. K. "Nonaveraging Sets. Nondividing Sets." §C16 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 131-132,
1994.Sloane, N. J. A. Sequence A051014
in "The On-Line Encyclopedia of Integer Sequences."Straus,
E. G. "Non-Averaging Sets." Proc. Symp. Pure Math19,
215-222, 1971.
is dividing, since 
(and 
), but 
is nondividing since 4 divides none of 
, and similarly for 6 and 7. The empty
set and sets of length one are therefore trivially nondividing.
Also, any set other than 
which contains 1 is dividing.
.
Then the numbers of nondividing subsets on 
, 
, ... are 1, 2, 3, 5, 7, 11, 14, 21, ... (OEIS A051014).
For example, the 14 nondividing sets in 
are 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
.
