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Semi r-Free and r-Free Integers—A Unified Approach

Published online by Cambridge University Press:  20 November 2018

G. E. Hardy  and
M. V. Subbarao
G. E. Hardy
Affiliation:
University of Alberta Edmonton, Alberta. T6G 2G1
M. V. Subbarao
Affiliation:
University of Alberta Edmonton, Alberta. T6G 2G1
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Abstract

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We obtain an asymptotic formula for the number of (k, r)-free integers that do not exceed x. By definition, a (k, r)-free integer is one in whose canonical representation no prime power is in the interval [r, k−1] where 1 < r < k are fixed integers. These include as special cases the r-free integers, the semi r-free integers and the k-full integers. We obtain an asymptotic formula for the number of representations of an integer as the sum of a prime and a (k, r)-free integer, and use the result to prove that every sufficiently large integer can be represented as the sum of a prime and m = abk where a and b are both square free, (a, b) = 1, b > 1 and k is any fixed integer, k ≥ 3.

Keywords

10A20r-free numberssemi r-free numbersRiemann zeta functionMöbius function
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 3 , 01 September 1982 , pp. 273 - 290
Copyright
Copyright © Canadian Mathematical Society 1982

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