Type: Article

On the index of composition of the Euler function and of the sum of divisors function

Jean-Marie De Koninck; Florian Luca;

Abstract:

Given an integer n >= 2, let lambda(n) := (log n)/(log gamma(n)), where gamma(n) = Pi(p vertical bar n) p, denote the index of composition of n, with lambda(1) = 1. Letting phi and sigma stand for the Euler function and the sum of divisors function, we show that both lambda(phi(n)) and; lambda(sigma(n)) have normal order 1 and mean value 1. Given an arbitrary integer k >= 2, we then study the size of min {lambda(phi(n)), lambda(phi(n + 1)), ..., lambda(phi(n + k - 1))} and of min {lambda(sigma(n)), lambda(sigma(n + 1)), ..., lambda(sigma(n + k - 1))} as n becomes large.