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New results on the least common multiple of consecutive integers

When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions g k (k ∈ N), defined by g k (n):= n(n+1)...(n+k) lcm(n,n+1,...,n+k) (Vn ∈ N \ {0}). He proved that for each k ∈ N, g k is periodic and k! is a period of g k . He raised the open problem of determining the smallest positive period P k of g k . Very recently, S. Hong and Y. Yang improved the period k! of g k to lcm(1,2,..., k). In addition, they conjectured that P k is always a multiple of the positive integer lcm(1,2,...,k,k+1) k+1. An immediate consequence of this conjecture is that if (k+1) is prime, then the exact period of g k is precisely equal to lcm(1, 2 k). In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of P k (k ∈ N). We deduce, as a corollary, that P k is equal to the part of lcm(1, 2 k) not divisible by some prime.