On Bounds of Non-Deficient Numbers

Abstract

Objectives: To improve the upper bounds of a quasi perfect number and give an important result on its divisibility with primes. Methods: A positive integer n is quasi perfect if s (n) >2n + 1, where s (n) denotes the sum of the positive divisors of n. However, the existence of a quasi perfect number, which is a Non-Deficient number, is still an open problem. We use R(n), the sum of the reciprocals of distinct primes dividing the quasi perfect number, to derive lemmas and improve the bounds obtained by earlier authors. Findings: We improve the upper bounds for R(n), when n is quasi perfect with gcd (15, n) = 3 or gcd (15, n) = 5. As a consequence, we establish that a quasi perfect number, if exists, is divisible by both 3 and 5 or by none of them. Novelty: The unique method of using R(n) also resulted in finding an important result that 3, 5 and 7 cannot divide any quasi perfect number. Mathematics Subject Classification: 11A05, 11A25. Keywords: non-deficient number; quasi perfect number; sum of the divisor; sum of the reciprocal; bounds of perfect number; number of divisors