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Sequences which omit powers

Published online by Cambridge University Press:  01 August 2016

R. D. Nelson
R. D. Nelson*
Affiliation:
Department of Education, University of Manchester, M13 9PL
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Extract

Here is an interesting result on sequences amenable to investigation and proof in the classroom. Consider the sequence

where, as usual, [x] denotes the integer part of x. You will find that you get the sequence

2, 3, 5, 6, 7, 8, 10, . . .

which consists of the natural numbers with the perfect squares omitted. This result is due to H. Halberstam [1] and in this article we verify it and look for generalisations to sequences which omit mth powers.

Type
Research Article
Information
The Mathematical Gazette , Volume 72 , Issue 461 , October 1988 , pp. 208 - 211
Copyright
Copyright © The Mathematical Association 1988

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References

1. Halberstam, H., “In praise of arithmetic”, in The teaching of algebra at the pre-college level edited by Braunfield, P. and Deskins, W.E., CEMREL, St. Louis (1975).Google Scholar