On quotients of values of Euler's function on the Catalan numbers

Abstract

© 2016 Elsevier Inc. In a recent work, Luca and Stănică examined quotients of the form φ(Cm)φ(Cn), where φ is Euler's totient function and C0,C1,C2… is the sequence of the Catalan numbers. They observed that the number 4 (and analogously 14) appears noticeably often as a value of these quotients. We give an explanation of this phenomenon, based on Dickson's conjecture. It turns out not only that the value 4 is (in a certain sense) special in relation to the quotients φ(Cn+1)φ(Cn), but also that the value 4k has similar “special” properties with respect to the quotients φ(Cn+k)φ(Cn), and in particular we show that Dickson's conjecture implies that, for each k, the number 4k appears infinitely often as a value of the quotients φ(Cn+k)φ(Cn).