For any positive integer let ϕ(n) be the Euler function of n. A positive integer is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression consists entirely of noncototients. We then use computations to detect seven such positive integers k.
We study the properties of the function which determines the number of representations of an integer as a sum of distinct Fibonacci numbers . We determine the maximum and mean values of for .
We study the properties of the function R(n) which determines the number of representations of an integer n as a sum of distinct Fibonacci numbers Fk. We determine the maximum and mean values of R(n) for Fk ≤ n < Fk+1.
Integrals of logarithmic and hypergeometric functions are intrinsically connected with Euler sums. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.