Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T16:19:31.028Z Has data issue: false hasContentIssue false

BOUNDS FOR ODD k-PERFECT NUMBERS

Published online by Cambridge University Press:  21 July 2011

SHI-CHAO CHEN*
Affiliation:
Institute of Applied Mathematics, School of Mathematics and Information Science, Henan University, Kaifeng 475004, PR China (email: [email protected])
HAO LUO
Affiliation:
Institute of Applied Mathematics, School of Mathematics and Information Science, Henan University, Kaifeng 475004, PR China (email: [email protected])
*
For correspondence; e-mail: [email protected]
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.

Let k≥2 be an integer. A natural number n is called k-perfect if σ(n)=kn. For any integer r≥1, we prove that the number of odd k-perfect numbers with at most r distinct prime factors is bounded by (k−1)4r3.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

Supported by the Natural Science Foundation of China (Grant 11026080) and the Natural Science Foundation of Education Department of Henan Province (Grant 2009A110001).

References

[1]Cohen, G. L. and Hagis, P. Jr, ‘Results concerning odd multiperfect numbers’, Bull. Malays. Math. Soc. 8 (1985), 2326.Google Scholar
[2]Cohen, G. L. and Hendy, M. D., ‘Polygonal supports for sequences of primes’, Math. Chronicle 9 (1980), 120136.Google Scholar
[3]Cook, R. J., ‘Bounds for odd perfect numbers’, in: Number Theory (Ottawa, ON, 1996), CRM Proceedings & Lecture Notes, 19 (American Mathematical Society, Providence, RI, 1999), pp. 6771.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), 413422.Google Scholar
[5]Heath-Brown, D. R., ‘Odd perfect numbers’, Math. Proc. Cambridge Philos. Soc. 115 (1994), 191196.CrossRefGoogle Scholar
[6]Nielsen, P., ‘An upper bound for odd perfect numbers’, Integers 3 (2003), A14, 9pp (electronic).Google Scholar
[7]Nielsen, P., ‘Odd perfect numbers have at least nine distinct prime factors’, Math. Comp. 76 (2007), 21092126.Google Scholar
[8]Pollack, P., ‘On Dickson’s theorem concerning odd perfect numbers’, Amer. Math. Monthly 118 (2011), 161164.Google Scholar
[9]Pomerance, C., ‘Multiply perfect numbers, Mersenne primes and effective computability’, Math. Ann. 226 (1977), 195206.CrossRefGoogle Scholar
[10]Wirsing, E., ‘Bemerkung zu der Arbeit über vollkommene Zahlen’, Math. Ann. 137 (1959), 316318.CrossRefGoogle Scholar