Bi-unitary multiperfect numbers, I

Pentti Haukkanen and Varanasi Sitaramaiah
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 1, Pages 93–171
DOI: 10.7546/nntdm.2020.26.1.93-171
Full paper (PDF, 432 Kb)

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Authors and affiliations

Pentti Haukkanen
Faculty of Information Technology and Communication Sciences
FI-33014 Tampere University, Finland

Varanasi Sitaramaiah
1/194e, Poola Subbaiah Street, Taluk Office Road, Markapur
Prakasam District, Andhra Pradesh, 523316 India

Abstract

A divisor d of a positive integer n is called a unitary divisor if gcd(d, n/d) = 1; and d is called a bi-unitary divisor of n if the greatest common unitary divisor of d and n/d is unity. The concept of a bi-unitary divisor is due to D. Surynarayana [12]. Let σ^∗∗(n) denote the sum of the bi-unitary divisors of n. A positive integer n is called a bi-unitary perfect number if σ^∗∗(n) = 2n. This concept was introduced by C. R. Wall in 1972 [15], and he proved that there are only three bi-unitary perfect numbers, namely 6, 60 and 90.
In 1987, Peter Hagis [6] introduced the notion of bi-unitary multi k-perfect numbers as solutions n of the equation σ^∗∗(n) = kn. A bi-unitary multi 3-perfect number is called a bi-unitary triperfect number. A bi-unitary multiperfect number means a bi-unitary multi k-perfect number with k ≥ 3. Hagis [6] proved that there are no odd bi-unitary multiperfect numbers. We aim to publish a series of papers on bi-unitary multiperfect numbers focusing on multiperfect numbers of the form n = 2^au, where u is odd. In this paper–part I of the series–we investigate bi-unitary triperfect numbers of the form n = 2^au, where 1 ≤ a ≤ 3. It appears that n = 120 = 2³15 is the only such number. Hagis [6] found by computer the bi-unitary multiperfect numbers less than 10⁷. We have found 31 such numbers up to 8.10¹⁰. The first 13 are due to Hagis. After completing this paper we noticed that further numbers are already listed in The On-Line Encyclopedia of Integer Sequences (sequence A189000 Bi-unitary multiperfect numbers). The numbers listed there have been found by direct computer calculations. Our purpose is to present a mathematical search and treatment of bi-unitary multiperfect numbers.

Keywords

• Perfect numbers
• Triperfect numbers
• Multiperfect numbers
• Bi-unitary analogues

2010 Mathematics Subject Classification

• 11A25

References

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