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On Some Sequences Related to Sums of Powers
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Robert Dawson

Dept. of Mathematics and Computing Science

Saint Mary’s University

Halifax, NS B3L 3C3

Canada

**Abstract:**

Automorphic numbers (in a specified base) have the property that
the expansion of *n*^{2} 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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