On the Largest Integer that is not a Sum of Distinct Positive nth Powers
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(xn) = (1n,
2n, ...) is complete, meaning that every
sufficiently large integer is a sum of distinct nth powers of
positive integers. We prove that every integer
m ≥
(b - 1)2n-1(r + (2/3)(b - 1)(22n - 1) + 2(b - 2))n - 2a + ab,
where
a = n!2n2,
b= 2n3an-1,
r = 2n2 - na, is a sum of
distinct positive nth 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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