Authors’ summary: When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions , defined by
He proved that for each , is periodic and is a period of . He raised the open problem of determining the smallest positive period of . Very recently, S. Hong and Y. Yang [C. R., Math., Acad. Sci. Paris 346, No. 13–14, 717–721 (2008; Zbl 1213.11014)] improved the period of to . In addition, they conjectured that is always a multiple of the positive integer . An immediate consequence of this conjecture is that if is prime, then the exact period of is precisely equal to .
In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of . We deduce, as a corollary, that is equal to the part of not divisible by some prime.