Pentti Haukkanen and Varanasi Sitaramaiah

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 1, Pages 45—69

DOI: 10.7546/nntdm.2021.27.1.45-69

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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 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(b) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we considered bi-unitary triperfect numbers of the form , where and is odd. In part IV(a) we solved partly the case . We proved that if is a bi-unitary triperfect number of the form , where , then . We then solved completely the case . In the present paper we give some partial results concerning the case under the assumption

### 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 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.
- 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(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, 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.

## Cite this paper

Haukkanen, P., & Sitaramaiah, V. (2021). Bi-unitary multiperfect numbers, IV(b). Notes on Number Theory and Discrete Mathematics, 27(1), 45–69, doi: 10.7546/nntdm.2021.27.1.45-69.