Bi-unitary multiperfect numbers, II

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)


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


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 \sigma^{**}(n) denote the sum of the bi-unitary divisors of n. A positive integer n is called a bi-unitary multiperfect number if \sigma^{**}(n)=kn for some k\geq 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 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 n=2^{a}u, where 1\leq a \leq 3 and u is odd; the only one being n=120. In this second part we find all bi-unitary triperfect numbers in the cases a=4 and a=5. For a=4 the only one is n=2160, and for a=5 they are n=672, n=10080, n=528800 and n=22932000.


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

2010 Mathematics Subject Classification

  • 11A25


  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 Number Theory Discrete Math., 26 (1), 93–171.
  3. Sándor, J. & Crstici, P. (2004). Handbook of Number Theory II, Kluwer Academic.
  4. 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.
  5. Wall, C. R. (1972). Bi-unitary perfect numbers, Proc. Amer. Math. Soc., 33, No. 1, 39–42.

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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.

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