Pentti Haukkanen and Varanasi Sitaramaiah

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 3, Pages 33—67

DOI: 10.7546/nntdm.2020.26.3.33-67

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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 (1972). Let *σ*^{∗∗}(*n*) denote the sum of the bi-unitary divisors of *n*. A positive integer *n* is called a bi-unitary multiperfect number if *σ*^{∗∗}(*n*) = *kn* for some *k* ≥ 3. For *k* = 3 we obtain the bi-unitary triperfect numbers.

Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is part III in a series of papers on even bi-unitary multiperfect numbers. In parts I and II we found all bi-unitary triperfect numbers of the form *n* = 2* ^{a}u*, where 1 ≤

*a*≤ 5 and

*u*is odd. There exist exactly six such numbers. In this part we examine the case

*a*= 6. We prove that if

*n*= 2

^{6}

*u*is a bi-unitary triperfect number, then

*n*= 22848,

*n*= 342720,

*n*= 51979200 or

*n*= 779688000.

### Keywords

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

### 2010 Mathematics Subject Classification

- 11A25

### References

- Hagis, P., Jr. (1987). Bi-unitary amicable and multiperfect numbers, Fibonacci Quart., 25 (2), 144–150.
- Haukkanen, P. & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, I, Notes Number Theory Discrete Math., 26 (1), 93–171.
- Haukkanen, P., & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, II. Notes on Number Theory and Discrete Mathematics, 26 (2), 1-26.
- Sándor, J. & Crstici, P. (2004). Handbook of Number Theory II, Kluwer Academic.
- Suryanarayana, D. (1972). The number of bi-unitary divisors of an integer, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics 251: 273–282, New York, Springer–Verlag.
- Wall, C. R. (1972). Bi-unitary perfect numbers, Proc. Amer. Math. Soc., 33, No. 1, 39–42.

## Related papers

- Haukkanen, P. & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, I, Notes Number Theory Discrete Math., 26 (1), 93–171, doi: 10.7546/nntdm.2020.26.1.93-171.
- Haukkanen, P., & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, II. Notes on Number Theory and Discrete Mathematics, 26 (2), 1-26, doi: 10.7546/nntdm.2020.26.2.1-26.

## Cite this paper

Haukkanen, P., & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, III. Notes on Number Theory and Discrete Mathematics, 26 (3), 33-67, doi: 10.7546/nntdm.2020.26.3.33-67.