Bi-unitary multiperfect numbers, IV(c)

Pentti Haukkanen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 3, Pages 411–434
DOI: 10.7546/nntdm.2022.28.3.411-434
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Pentti Haukkanen
Faculty of Information Technology and Communication Sciences
FI-33014 Tampere University, Finland

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(c) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we determined all bi-unitary triperfect numbers of the form n=2^{a}u, where 1\leq a \leq 6 and u is odd. In part V we fixed the case a=8. The case a=7 is more difficult. In Parts IV(a-b) we solved partly this case, and in the present paper (Part IV(c)) we continue the study of the same case (a=7).

Keywords

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

2020 Mathematics Subject Classification

  • 11A25

References

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  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. Haukkanen, P., & Sitaramaiah, V. (2020). Bi-unitary multiperfect numbers, III. Notes on Number Theory and Discrete Mathematics, 26(3), 33–67.
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  6. Haukkanen, P., & Sitaramaiah, V. (2021). Bi-unitary multiperfect numbers, IV(b). Notes on Number Theory and Discrete Mathematics, 27(1), 45–69.
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Manuscript history

  • Received: 2 June 2022
  • Revised: 12 July 2022
  • Accepted: 13 July 2022
  • Online First: 13 July 2022

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Cite this paper

Haukkanen, P. (2022). Bi-unitary multiperfect numbers, IV(c). Notes on Number Theory and Discrete Mathematics, 28(3), 411-434, DOI: 10.7546/nntdm.2022.28.3.411-434.

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