**Pentti Haukkanen**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 28, 2022, Number 3, Pages 411–434

DOI: 10.7546/nntdm.2022.28.3.411-434

**Full paper (PDF, 287 Kb)**

## Details

### Authors and affiliations

Pentti Haukkanen

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

### 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 IV(c) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we determined all bi-unitary triperfect numbers of the form , where and is odd. In part V we fixed the case . The case is more difficult. In Parts IV(a-b) we solved partly this case, and in the present paper (Part IV(c)) we continue the study of the same case ().

### Keywords

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

### 2020 Mathematics Subject Classification

- 11A25

### References

- Hagis, P., Jr. (1987). Bi-unitary amicable and multiperfect numbers.
*The Fibonacci Quarterly*, 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. - 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. - Haukkanen, P., & Sitaramaiah, V. (2021). Bi-unitary multiperfect numbers, V.
*Notes on Number Theory and Discrete Mathematics*, 27(2), 20–40. - Sándor, J., & Atanassov, K. T. (2021).
*Arithmetic Functions*. Nova Science Publishers. - Sándor, J., & Crstici, P. (2004).
*Handbook of Number Theory, Vol. II*, Kluwer Academic. - Sitaramaiah, V. (2020). Personal communication.
- Suryanarayana, D. (1972). The number of bi-unitary divisors of an integer.
*The Theory of Arithmetic Functions*, Lecture Notes in Mathematics 251: 273–282, New York, Springer–Verlag. - Wall, C. R. (1972). Bi-unitary perfect numbers.
*Proceedings of the American Mathematical Society*, 33(1), 39–42.

### Manuscript history

- Received: 2 June 2022
- Revised: 12 July 2022
- Accepted: 13 July 2022
- Online First: 13 July 2022

## Related papers

## Cite this paper

Haukkanen, P. (2022). Bi-unitary multiperfect numbers, IV(c). *Notes on Number Theory and Discrete Mathematics*, 28(3), 411-434, DOI: 10.7546/nntdm.2022.28.3.411-434.