Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.7

Improved Bounds on the Anti-Waring Number

Paul LeVan and David Prier
Department of Mathematics
Gannon University
Erie, PA 16541-0001


The Anti-Waring number, N(k,r), is defined to be the least integer such that it and every larger integer can be written as the sum of the kth powers of r or more distinct positive integers. Several authors have examined this variation of the classical Waring problem. We provide improved bounds for N(k,r) in general and when k = 2. We then connect this problem to the theory of partitions. We use traditional counting arguments, as well as a generating function methodology that has not yet been applied to finding the Anti-Waring number.

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(Concerned with sequences A001661.)

Received March 20 2017; revised versions received August 9 2017; August 11 2017. Published in Journal of Integer Sequences, September 5 2017.

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