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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.
- Quasiperfect number
- Special factor
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Cite this paper
Siva Rama Prasad, V., & Sunitha, C. (2017). On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, 23(3), 73-78.