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

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## Details

### 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*) = 2*n*. 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* ^{a}u*, where

*u*is odd. In this paper—part I of the series—we investigate bi-unitary triperfect numbers of the form

*n*= 2

*, where 1 ≤*

^{a}u*a*≤ 3. It appears that

*n*= 120 = 2

^{3}15 is the only such number. Hagis [6] found by computer the bi-unitary multiperfect numbers less than 10

^{7}. We have found 31 such numbers up to 8.10

^{10}. 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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## Related papers

- Haukkanen, P., & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, II. Notes on Number Theory and Discrete Mathematics, 26(2), 1-26.
- Haukkanen, P., & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, III. Notes on Number Theory and Discrete Mathematics, 26(3), 33-67.
- Haukkanen, P., & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, IV(a). Notes on Number Theory and Discrete Mathematics, 26(4), 2-32.
- Haukkanen, P., & Sitaramaiah, V. (2021). Bi-unitary multiperfect numbers, IV(b). Notes on Number Theory and Discrete Mathematics, 27(1), 45–69.

## Cite this paper

Haukkanen P. & Sitaramaiah V. (2020). Bi-unitary multiperfect numbers, I. Notes on Number Theory and Discrete Mathematics, 26(1), 93-171, doi: 10.7546/nntdm.2020.26.1.93-171.