Bi-unitary multiperfect numbers, I

Pentti Haukkanen and Varanasi Sitaramaiah
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 1, Pages 93—171
DOI: 10.7546/nntdm.2020.26.1.93-171
Download full paper: PDF, 432 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 [12]. Let σ∗∗(n) denote the sum of the bi-unitary divisors of n. A positive integer n is called a bi-unitary perfect number if σ∗∗(n) = 2n. This concept was introduced by C. R. Wall in 1972 [15], and he proved that there are only three bi-unitary perfect numbers, namely 6, 60 and 90.
In 1987, Peter Hagis [6] introduced the notion of bi-unitary multi k-perfect numbers as solutions n of the equation σ∗∗(n) = kn. A bi-unitary multi 3-perfect number is called a bi-unitary triperfect number. A bi-unitary multiperfect number means a bi-unitary multi k-perfect number with k ≥ 3. Hagis [6] proved that there are no odd bi-unitary multiperfect numbers. We aim to publish a series of papers on bi-unitary multiperfect numbers focusing on multiperfect numbers of the form n = 2au, where u is odd. In this paper—part I of the series—we investigate bi-unitary triperfect numbers of the form n = 2au, where 1 ≤ a ≤ 3. It appears that n = 120 = 2315 is the only such number. Hagis [6] found by computer the bi-unitary multiperfect numbers less than 107. We have found 31 such numbers up to 8.1010. The first 13 are due to Hagis. After completing this paper we noticed that further numbers are already listed in The On-Line Encyclopedia of Integer Sequences (sequence A189000 Bi-unitary multiperfect numbers). The numbers listed there have been found by direct computer calculations. Our purpose is to present a mathematical search and treatment of bi-unitary multiperfect numbers.


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

2010 Mathematics Subject Classification

  • 11A25


  1. Cohen, E. (1960). Arithmetical functions associated with the unitary divisors of an integer, Math. Z., 74, 66–80.
  2. Dickson, L. E. (1919). History of the theory of numbers, Volume-I (Divisibility and Primality) AMS Chelsea Publishing, American Mathematical Society, Providence, Rhode Island, USA.
  3. Guy, R. K. (1981). Unsolved problems in number theory, Springer–Verlag.
  4. Harris, V. C. & Subbarao, M. V. (1974). Unitary multiperfect numbers (Abstract), Notices Amer. Math. Soc., 21, A435.
  5. Hagis, P., Jr. (1984). Lower bound for unitary multiperfect numbers, Fibonacci Quart., 22, 140–143.
  6. Hagis, P., Jr. (1987). Bi-unitary amicable and multiperfect numbers, Fibonacci Quart., 25 (2), 144–150.
  7. Ochem, P. & Rao, M. (2012). Odd perfect numbers are greater than 101500, Math. Comp., 81, 1869–1877.
  8. Sándor, J. & Crstici, P. (2004). Handbook of Number Theory II, Kluwer Academic.
  9. Sitaramaiah, V. & Subbarao, M. V. (1998). On Unitary Multiperfect Numbers, Nieuw Arch. Wiskunde, 16, 57–61.
  10. Subbarao, M. V. (1970). Are there infinity of unitary perfect numbers? Amer. Math. Monthly, 77, 389–390.
  11. Subbarao, M. V. & Warren, L. J. (1966). Unitary perfect numbers, Canad. Math. Bull. 9, 147–153.
  12. 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.
  13. Vaithyanathaswamy, R. (1931). The theory of multiplicative arithmetic functions, Trans. Amer. Math. Soc., 33, 579–662.
  14. Wall, C. R. (1969). A new unitary perfect number, Notices Amer. Math. Soc., 16, 825.
  15. Wall, C. R. (1972). Bi-unitary perfect numbers, Proc. Amer. Math. Soc., 33, No. 1, 39–42.
  16. Wall, C. R. (1975). The fifth unitary perfect number, Canad. Math. Bull., 18, 115–122.
  17. Wikipedia contributors. (2019, September 28). Multiply perfect number. In Wikipedia, The Free Encyclopedia. Retrieved 2020, March 5, from

Related papers

Cite this paper

Haukkanen P. & Sitaramaiah V. (2020). Bi-unitary multiperfect numbers, I. Notes on Number Theory and Discrete Mathematics, 26(1), 93-171, doi: 10.7546/nntdm.2020.26.1.93-171.

Comments are closed.