Bi-unitary multiperfect numbers, III

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)

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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^au, 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 = 2⁶u 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

1. Hagis, P., Jr. (1987). Bi-unitary amicable and multiperfect numbers, Fibonacci Quart., 25 (2), 144–150.
2. Haukkanen, P. & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, I, Notes on Number Theory and Discrete Mathematics, 26 (1), 93–171.
3. Haukkanen, P., & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, II. Notes on Number Theory and Discrete Mathematics, 26 (2), 1-26.
4. Sándor, J. & Crstici, P. (2004). Handbook of Number Theory II, Kluwer Academic.
5. 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.
6. Wall, C. R. (1972). Bi-unitary perfect numbers, Proc. Amer. Math. Soc., 33, No. 1, 39–42.