**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

**Full paper (PDF, 324 Kb)**

## 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 on Number Theory and Discrete Mathematics*, 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 on Number Theory and Discrete Mathematics*, 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. - 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, III. *Notes on Number Theory and Discrete Mathematics*, 26(3), 33-67, DOI: 10.7546/nntdm.2020.26.3.33-67.