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New results on the least common multiple of consecutive integers. (English) Zbl 1229.11007

Authors’ summary: When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions \(g_k\) \((k \in \mathbb{N})\), defined by \[ g_k(n) := \frac{n(n+1)\dots (n+k)} {\text{lcm}(n, n+1, \dots, n+k)}\qquad (\forall n \in \mathbb{N} \setminus \{0\}). \] He proved that for each \(k\in \mathbb{N}\), \(g_k\) 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 [C. R., Math., Acad. Sci. Paris 346, No. 13–14, 717–721 (2008; Zbl 1213.11014)] improved the period \(k!\) of \(g_k\) to \(\text{lcm}(1, 2, \dots, k)\). In addition, they conjectured that \(P_k\) is always a multiple of the positive integer \( \frac{\text{lcm}(1, 2, \dots, k, k+1)}{k+1}\). An immediate consequence of this conjecture is that if \((k + 1)\) is prime, then the exact period of \( g_k\) is precisely equal to \(\text{lcm}(1, 2, \dots, k)\).
In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of \(P_k\) \((k\in \mathbb{N})\). We deduce, as a corollary, that \(P_k\) is equal to the part of \(\text{lcm}(1, 2, \dots, k)\) not divisible by some prime.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11B83 Special sequences and polynomials

Citations:

Zbl 1213.11014
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References:

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