On a diophantine equation of Ayad and Kihel

Abstract

Let f(n) denote the number of relatively prime sets in {1; : : : ; n}. This is sequence A085945 in Sloane’s On-Line Encyclopedia of Integer Sequences. Motivated by a paper of Ayad and Kihel [1], we show that there are at most finitely many positive integers n such that f(n) is a perfect power of exponent > 1 of some other integer. We also show that the sequence {f(n)}n ≥ 1 is not holonomic; that is, it satisfies no recurrence relation of finite order with polynomial coefficients.

Quaestiones Mathematicae 35(2012), 235–243