تسجيل الدخول

تسجيل

Nondividing Set

المؤلف:  Abbott, H. L.

المصدر:  "Extremal Problems on Non-Averaging and Non-Dividing Sets." Pacific J. Math. 91

الجزء والصفحة:  ...

3-11-2020

1581

Nondividing Set

A set in which no element divides the sum of any nonempty subset of the other elements. For example, {2,3,5}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline1.gif" style="height:15px; width:47px" /> is dividing, since  (and ), but {4,6,7}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline4.gif" style="height:15px; width:47px" /> is nondividing since 4 divides none of {6,7,(6+7)}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline5.gif" style="height:15px; width:77px" />, and similarly for 6 and 7. The empty set and sets of length one are therefore trivially nondividing. Also, any set other than {1}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline6.gif" style="height:15px; width:17px" /> which contains 1 is dividing.

Consider all possible subsets on the integers {1,2,...,n}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline7.gif" style="height:15px; width:99px" />. 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 , {1}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline12.gif" style="height:15px; width:17px" />, {2}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline13.gif" style="height:15px; width:17px" />, {3}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline14.gif" style="height:15px; width:17px" />, {4}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline15.gif" style="height:15px; width:17px" />, {5}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline16.gif" style="height:15px; width:17px" />, {6}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline17.gif" style="height:15px; width:17px" />, {2,3}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline18.gif" style="height:15px; width:32px" />, {2,5}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline19.gif" style="height:15px; width:32px" />, {3,4}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline20.gif" style="height:15px; width:32px" />, {3,5}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline21.gif" style="height:15px; width:32px" />, {4,5}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline22.gif" style="height:15px; width:32px" />, {4,6}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline23.gif" style="height:15px; width:32px" />, and {5,6}" src="https://mathworld.wolfram.com/images/equations/NondividingSet/Inline24.gif" style="height:15px; width:32px" />.

---

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