Nondividing Set

A set 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 on the integers . Then the numbers of nondividing subsets on , , ... are 1, 2, 3, 5, 7, 11, 14, 21, ... (OEIS A051014). For example, the 11 nondividing sets in  are , , , , , , , , , , , , , and .

REFERENCES:

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 Math 19, 215-222, 1971.a