Bi-unitary multiperfect numbers, IV(b)

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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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 \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 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 n=2^{a}u, where 1\leq a \leq 6 and u is odd. In part IV(a) we solved partly the case a=7. We proved that if n is a bi-unitary triperfect number of the form n=2^{7}.5^{b}.17^{c}.v, where (v, 2.5.17)=1, then b\geq 2. We then solved completely the case b=2. In the present paper we give some partial results concerning the case b\ge 3 under the assumption 3\nmid n.

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 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. Haukkanen, P., & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, III. Notes on Number Theory and Discrete Mathematics, 26(3), 33–67.
  5. Haukkanen, P., & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, IV(a). Notes on Number Theory and Discrete Mathematics, 26(4), 2–32.
  6. Sándor, J. & Crstici, P. (2004). Handbook of Number Theory II, Kluwer Academic.
  7. 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.
  8. Wall, C. R. (1972). Bi-unitary perfect numbers, Proc. Amer. Math. Soc., 33(1), 39–42.

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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.

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