Journal of Integer Sequences, Vol. 21 (2018), Article 18.7.6

On Some Sequences Related to Sums of Powers

Robert Dawson
Dept. of Mathematics and Computing Science
Saint Mary’s University
Halifax, NS B3L 3C3


Automorphic numbers (in a specified base) have the property that the expansion of n2 ends in that of n; Fairbairn characterized these numbers for all bases in 1969. Here we consider some related sequences: those n for which the sum of the first n natural numbers, squares, or cubes ends in n. For sums of natural numbers, these are Trigg's "trimorphic" numbers; for sums of squares, Pickover's "square pyramorphic" numbers. We characterize the trimorphic numbers for all bases, and the other two for base 10 and prime powers. We also solve a related problem due to Pickover.

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(Concerned with sequences A000124 A000217 A000330 A000537 A003226 A007185 A016090 A033819 A067270 A093534 A301912.)

Received April 4 2018; revised versions received July 19 2018; August 1 2018; September 7 2018. Published in Journal of Integer Sequences, September 8 2018.

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