Journal of Integer Sequences, Vol. 20 (2017), Article 17.7.5

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


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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