Nondividing Set

A set in which no element divides the sum of any nonempty subset of the other elements. For example, {2,3,5} is dividing, since 2|(3+5) (and 5|(2+3)), but {4,6,7} is nondividing since 4 divides none of {6,7,(6+7)}, and similarly for 6 and 7. The empty set and sets of length one are therefore trivially nondividing. Also, any set other than {1} which contains 1 is dividing.

Consider all possible subsets on the integers S_n={1,2,...,n}. Then the numbers of nondividing subsets on S_0, S_1, ... are 1, 2, 3, 5, 7, 11, 14, 21, ... (OEIS A051014). For example, the 11 nondividing sets in S_6 are emptyset, {1}, {2}, {3}, {4}, {5}, {6}, {2,3}, {2,5}, {3,4}, {3,5}, {4,5}, {4,6}, and {5,6}.

See also

Nonaveraging Sequence, Primitive Sequence

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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.

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Nondividing Set

Cite this as:

Weisstein, Eric W. "Nondividing Set." From MathWorld--A Wolfram Web Resource.

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