Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T09:48:45.503Z Has data issue: false hasContentIssue false
  • English
  • Français

NOTE ON ODD MULTIPERFECT NUMBERS

Part of: Elementary number theory Combinatorics

Published online by Cambridge University Press:  17 September 2012

LI-XIA DAI ,
HAO PAN  and
CUI-E TANG
LI-XIA DAI
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, PR China (email: lilidainjnu@163.com)
HAO PAN*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China (email: haopan79@yahoo.com.cn)
CUI-E TANG
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, PR China (email: tangcuie2008@126.com)
*
✉For correspondence; e-mail: haopan79@yahoo.com.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For $k\geq 2$ and $r\geq 1$, we prove that the number of odd $k$-perfect numbers with $r$ distinct prime factors is at most $4^{r^2}(k-1)^{2r^2+3}$.

Keywords

multiperfect numbermultiplicative partition function

MSC classification

Secondary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors 05A17: Partitions of integers
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 3 , June 2013 , pp. 448 - 451
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

References

[1]Chen, S.-C. & Luo, H., ‘Bounds for odd $k$-perfect numbers’, Bull. Aust. Math. Soc. 84(3) (2011), 475–480.CrossRefGoogle Scholar
[2]Chen, W.-L., ‘Upper bound of the number of multiplicative partitions’, Acta Math. Sinica (Chin. Ser.) 32 (1989), 604–609 (in Chinese).Google Scholar
[3]Cook, R. J., ‘Bounds for odd perfect numbers’, in: Number Theory (Ottawa, ON, 1996), 67–71, CRM Proc. Lecture Notes, 19 (American Mathematical Society, Providence, RI, 1999).Google Scholar
[4]Dickson, L. E., ‘Finiteness of the odd perfect and primitive abundant numbers with $n$ distinct prime factors’, Amer. J. Math. 35 (1913), 413–422.CrossRefGoogle Scholar
[5]Heath-Brown, D. R., ‘Odd perfect numbers’, Math. Proc. Cambridge Philos. Soc. 115 (1994), 191–196.CrossRefGoogle Scholar
[6]McCarthy, P. J., ‘Note on perfect and multiply perfect numbers’, Portugal Math. 16 (1957), 19–21.Google Scholar
[7]Nielsen, P. P., ‘An upper bound for odd perfect numbers’, Integers 3 (2003), A14, 9pp (electronic).Google Scholar
[8]Pollack, P., ‘On Dickson’s theorem concerning odd perfect numbers’, Amer. Math. Monthly 118 (2011), 161–164.CrossRefGoogle Scholar
[9]Pomerance, C., ‘Multiply perfect numbers, Mersenne primes, and effective computability’, Math. Ann. 226 (1977), 195–206.CrossRefGoogle Scholar