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On k-Imperfect Numbers
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Weiyi Zhou
Published/Copyright:
May 7, 2009
Abstract
A positive integer n is called a k-imperfect number if kρ(n) = n for some integer k ⩾ 2, where ρ is a multiplicative arithmetic function defined by ρ(pa) = pa – pa–1 + pa–2 – ⋯ + (–1)a for a prime power pa. In this paper, we prove that every odd k-imperfect number greater than 1 must be divisible by a prime greater than 102, give all k-imperfect numbers less than 232 = 4 294 967 296, and give several necessary conditions for the existence of an odd k-imperfect number.
Keywords.: k-imperfect number; cyclotomic polynomial
Received: 2008-05-30
Accepted: 2008-12-22
Published Online: 2009-05-07
Published in Print: 2009-April
© de Gruyter 2009
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Articles in the same Issue
- On k-Imperfect Numbers
- On Universal Binary Hermitian Forms
- The Shortest Game of Chinese Checkers and Related Problems
- On Two-Point Configurations in a Random Set
- On Rapid Generation of SL2(𝔽q)
- Tiling Proofs of Some Formulas for the Pell Numbers of Odd Index
- Some new van der Waerden numbers and some van der Waerden-type numbers
- Integer Partitions into Arithmetic Progressions with an Odd Common Difference
- On Asymptotic Constants Related to Products of Bernoulli Numbers and Factorials
