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
Full paper (PDF, 174 Kb)

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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

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