• Compositions with the Euler and Carmichael Functions 

    Banks, William David, 1964-; Luca, Florian; Saidak, Filip; Stănică, Pantelimon (Springer Verlag, 2005)
    Let ' and _ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that '(_(n)) = _('(n)). We also study the normal order of ...
  • Values of Arithmetical Functions Equal to a Sum of Two Squares 

    Banks, William David, 1964-; Luca, Florian; Saidak, Filip; Shparlinski, Igor E. (Oxford University Press, 2005)
    Let '(n) denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n ≤ x for which '(n) is the sum of two square numbers. We also obtain similar results for the Dedekind ...