On n Satisfying, n Dividing φ(n)σ(n)+1

Abstract

Let denotes the sum of the positive divisors of the positive integer and be the Euler’s totient function([1],page25). Cleary, divides if is a prime. Then, the question is there a composite that divides ? Considering this problem Yang-Gao Chen and Jin-Hui Fang [5] have proved that , where as usual is the number of distinct prime factors of. and , .We devoted the study of this problem where we prove that every in is odd and that , from which it follows that for any composite in the least prime factors is . Also, we obtained lower bounds for and for any with , which improves the result of [5] in some cases