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Improved Bounds on the Anti-Waring Number
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Paul LeVan and David Prier

Department of Mathematics

Gannon University

Erie, PA 16541-0001

USA

**Abstract:**

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 *k*th 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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