On the prime factors of a quasiperfect number

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
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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

Abstract

A positive integer N is said to be quasiperfect if σ(N) = 2N + 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

  1. Abbott, H. L., Aull, C. E., Brown, E. & Suryanarayana, D. (1976). Quasiperfect numbers,Acta Arithmetica, XXII(1973),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. & Hagis Jr., P. (1982). Some results concerning quasiperfect numbers, J. Austral. Math. Soc. (Ser.A), 33, 275–286.
  4. Kishore, M. (1975). Quasiperfect numbers are divisible by at least six distinct divisors, Notices. AMS, 22, A441.
  5. Sandor, J. & Crstici, B. (2004). Hand book of Number Theory II, Kluwer Academic Publishers, Dordrecht/Boston/London.
  6. Sierpinski, W. (1964). A Selection of problems in the Theory of Numbers, New York, (see page 110).
  7. Siva Rama Prasad, V. & Sunitha, C. (2017). On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, 23 (3), 73–78.

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  1. Siva Rama Prasad, V. & Sunitha, C. (2017). On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, 23 (3), 73–78.
  2. 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

APA

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.

Chicago

Siva Rama Prasad, V. and C. Sunitha. “On the prime factors of a quasiperfect number.” Notes on Number Theory and Discrete Mathematics 25, no. 2 (2019): 16-21, doi: 10.7546/nntdm.2019.25.2.16-21.

MLA

Siva Rama Prasad, V. and C. Sunitha. “On the prime factors of a quasiperfect number.” Notes on Number Theory and Discrete Mathematics 25.2 (2019): 16-21. Print, doi: 10.7546/nntdm.2019.25.2.16-21.

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