V. Siva Rama Prasad and C. Sunitha
Notes on Number Theory and Discrete Mathematics, ISSN 13105132
Volume 23, 2017, Number 3, Pages 73—78
Download full paper: PDF, 158 Kb
Details
Authors and affiliations
V. Siva Rama Prasad
Nalla Malla Reddy Engineering College, Divyanagar
Ghatakesar Mandal, Ranga Reddy District,Telangana501301, India
C. Sunitha
Department of Mathematics and Statistics, RBVRR Women’s College
Narayanaguda, Hyderabad, Telangana500027, 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
 Abbott, H. L., C. E. Aull, Brown, E., & D.Suryanarayana (1973) Quasiperfect numbers, Acta Arithmetica, XXII, 439447; correction to the paper, Acta Arithmetica, XXIX (1976), 636–637.
 Cattaneo, P. (1951), Sui numeri quasiperfetti, Boll. Un. Mat. Ital., 6(3), 59–62.
 Cohen, G. L. (1982) The nonexistence of quasiperfect numbers of certain form, Fib. Quart., 20(1), 81–84.
 Cohen, G. L. & Peter Hagis Jr. (1982) Some results concerning quasiperfect numbers, J.Austral.Math.Soc.(Ser.A), 33, 275–286.
 Kishore, M. (1975) Quasiperfect numbers are divisible by at least six distinct divisors, Notices. AMS, 22, A441.
 Sandor , J. & Crstici, B. (2004) Hand book of Number Theory II, Kluwer Academic Publishers, Dordrecht/ Boston/ London.
 Sierpinski, W. A Selection of problems in the Theory of Numbers, New York, (page 110).
Related papers
 Siva Rama Prasad, V., & Sunitha, C. (2019). On the prime factors of a quasiperfect number. Notes on Number Theory and Discrete Mathematics, 25 (2), 1621.

Reddy, P. A., Sunitha, C. & Prasad, V. Siva Rama. (2020). On quasimultiperfect numbers. Notes on Number Theory and Discrete Mathematics, 26 (3), 6873.
Cite this paper
APASiva Rama Prasad, V., & Sunitha, C. (2017). On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, 23(3), 7378.
ChicagoSiva Rama Prasad, V., and C. Sunitha. “On quasiperfect numbers.” Notes on Number Theory and Discrete Mathematics 23, no. 3 (2017): 7378.
MLASiva Rama Prasad, V., and C. Sunitha. “On quasiperfect numbers.” Notes on Number Theory and Discrete Mathematics 23.3 (2017): 7378. Print.