Pentti Haukkanen and Varanasi Sitaramaiah

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 2, Pages 1–26

DOI: 10.7546/nntdm.2020.26.2.1-26

**Full paper (PDF, 258 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 of a positive integer is called a unitary divisor if and is called a bi-unitary divisor of if the greatest common unitary divisor of and is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let denote the sum of the bi-unitary divisors of . A positive integer is called a bi-unitary multiperfect number if for some . For 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 II in a series of papers on even bi-unitary multiperfect numbers. In the first part we found all bi-unitary triperfect numbers of the form , where and is odd; the only one being . In this second part we find all bi-unitary triperfect numbers in the cases and . For the only one is , and for they are , , and .

### 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.
- 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, 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, II. *Notes on Number Theory and Discrete Mathematics*, 26 (2), 1-26, DOI: 10.7546/nntdm.2020.26.2.1-26.