V. Siva Rama Prasad and C. Sunitha

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 2, Pages 16–21

DOI: 10.7546/nntdm.2019.25.2.16-21

**Full paper (PDF, 174 Kb)**

## Details

### Authors and affiliations

V. Siva Rama Prasad

*Nalla Malla Reddy Engineering College,
Divyanagar,Ghatkesar Mandal, Ranga Reddy District,
Telangana-501301, India
*

C. Sunitha

*Department of Mathematics and Statistics,*

RBVRR Women’s College, Narayanaguda, Hyderabad,

Telangana-500027, India

RBVRR Women’s College, Narayanaguda, Hyderabad,

Telangana-500027, India

### Abstract

A positive integer *N* is said to be *quasiperfect *if *σ*(*N*) = 2*N* + 1 where *σ*(*N*) is the sum of the positive divisors of *N*. So far no quasiperfect number is known. If such *N* exists, let *γ*(*N*) denote the product of the distinct primes dividing *N*. In this paper, we obtain a lower bound for *γ*(*N*) in terms of *r* = *ω*(*N*), the number of distinct prime factors of *N*. Also we show that every quasiperfect number *N* is divisible by a prime *p* with (i) *p* ≡ 1 (mod 4); (ii) *p* ≡ 1 (mod 5) if 5 ∤ *N* and (iii) *p* ≡ 1 (mod 3), if 3 ∤ *N*.

### Keywords

- Quasiperfect number
- Radical of an integer

### 2010 Mathematics Subject Classification

- 05C15

### References

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- Cattaneo, P. (1951). Sui numeri quasiperfetti, Boll. Un. Mat. Ital., 6 (3), 59–62.
- Cohen G. L. & Hagis Jr., P. (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.
- Sándor, J. & Crstici, B. (2004). Hand book of Number Theory II, Kluwer Academic Publishers, Dordrecht/Boston/London.
- Sierpinski, W. (1964). A Selection of problems in the Theory of Numbers, New York, (see page 110).
- Siva Rama Prasad, V. & Sunitha, C. (2017). On quasiperfect numbers.
*Notes on Number Theory and Discrete Mathematics*, 23 (3), 73–78.

## Related papers

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*Notes on Number Theory and Discrete Mathematics*, 23 (3), 73–78. - Reddy, P. A., Sunitha, C. & Prasad, V. Siva Rama. (2020). On quasimultiperfect numbers.
*Notes on Number Theory and Discrete Mathematics*, 26 (3), 68-73.

## Cite this paper

Siva Rama Prasad, V. & Sunitha, C. (2019). On the prime factors of a quasiperfect number. *Notes on Number Theory and Discrete Mathematics*, 25(2), 16-21, DOI: 10.7546/nntdm.2019.25.2.16-21.