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On the Largest Integer that is not a Sum of Distinct Positive ***n*th Powers

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Doyon Kim

Department of Mathematics and Statistics

Auburn University

Auburn, AL 36849

USA

**Abstract:**

It is known that for an arbitrary positive integer *n* the sequence
*S*(*x*^{n}) = (1^{n},
2^{n}, ...) is complete, meaning that every
sufficiently large integer is a sum of distinct *n*th powers of
positive integers. We prove that every integer

*m* ≥
(*b* - 1)2^{n-1}(*r* + (2/3)(*b* - 1)(2^{2n} - 1) + 2(*b* - 2))^{n} - 2*a* + *ab*,

where
*a* = *n*!2^{n2},
*b*= 2^{n3}*a*^{n-1},
*r* = 2^{n2 - n}*a*, is a sum of
distinct positive *n*th powers.

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(Concerned with sequence
A001661.)

Received October 29 2016; revised versions received November 2 2016;
July 1 2017. Published in *Journal of Integer Sequences*,
July 2 2017.

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