On quasiperfect numbers

V. Siva Rama Prasad and C. Sunitha
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 23, 2017, Number 3, Pages 73–78
Full paper (PDF, 158 Kb)

Details

Authors and affiliations

V. Siva Rama Prasad
Nalla Malla Reddy Engineering College, Divyanagar
Ghatakesar Mandal, Ranga Reddy District,Telangana-501301, India

C. Sunitha
Department of Mathematics and Statistics, RBVRR Women’s College
Narayanaguda, Hyderabad, Telangana-500027, India

Abstract

A natural number N is said to be quasiperfect if σ(N) = 2N + 1 where σ(N) is the sum of the positive divisors of N. No quasiperfect number is known. If a quasiperfect number N exists and if ω(N) is the number of distinct prime factors of N then G. L. Cohen has proved ω(N) ≥ 7 while H. L. Abbott et. al have shown ω(N) ≥ 10 if (N, 15) = 1. In this paper we first prove that every quasiperfect numbers N has an odd number of special factors (see definition 2.3 below) and use it to show that ω(N) ≥ 15 if (N, 15) = 1 which refines the result of Abbott et.al. Also we provide an alternate proof of Cohen’s result when (N, 15) = 5.

Keywords

• Quasiperfect number
• Special factor

AMS Classification

• 11A25

References

1. Abbott, H. L., C. E. Aull, Brown, E., & D.Suryanarayana (1973) Quasiperfect numbers, Acta Arithmetica, XXII, 439-447; correction to the paper, Acta Arithmetica, XXIX (1976), 636–637.
2. Cattaneo, P. (1951), Sui numeri quasiperfetti, Boll. Un. Mat. Ital., 6(3), 59–62.
3. Cohen, G. L. (1982) The non-existence of quasiperfect numbers of certain form, Fib. Quart., 20(1), 81–84.
4. Cohen, G. L. & Peter Hagis Jr. (1982) Some results concerning quasiperfect numbers, J.Austral.Math.Soc.(Ser.A), 33, 275–286.
5. Kishore, M. (1975) Quasiperfect numbers are divisible by at least six distinct divisors, Notices. AMS, 22, A441.
6. Sándor , J. & Crstici, B. (2004) Hand book of Number Theory II, Kluwer Academic Publishers, Dordrecht/ Boston/ London.
7. Sierpinski, W. A Selection of problems in the Theory of Numbers, New York, (page 110).