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

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## 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*) = 2*N* + 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, 439-447; 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 non-existence 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).

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## Cite this paper

APASiva Rama Prasad, V., & Sunitha, C. (2017). On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, 23(3), 73-78.

ChicagoSiva Rama Prasad, V., and C. Sunitha. “On quasiperfect numbers.” Notes on Number Theory and Discrete Mathematics 23, no. 3 (2017): 73-78.

MLASiva Rama Prasad, V., and C. Sunitha. “On quasiperfect numbers.” Notes on Number Theory and Discrete Mathematics 23.3 (2017): 73-78. Print.