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We present here a method which allows to derive a nontrivial lower bounds forthe least common multiple of some finite sequences of integers. We obtainefficient lower bounds (which in a way are optimal) for the arithmeticprogressions and lower bounds less efficient (but nontrivial) for quadraticsequences whose general term has the form $u_n = a n (n + t) + b$ with $(a, t,b) \in {\mathbb{Z}}^3, a \geq 5, t \geq 0, \rm{gcd}(a, b) = 1$. From this, wededuce for instance the lower bound: $\mathrm{lcm}\{1^2 + 1, 2^2 + 1, ..., n^2+ 1\} \geq 0,32 (1,442)^n$ (for all $n \geq 1$).  In the last part of this article, we study the integer $\mathrm{lcm}(n, n +1, ..., n + k)$ $(k \in \mathbb{N}, n \in {\mathbb{N}}^*)$. We show that it hasa divisor $d_{n, k}$ simple in its dependence on $n$ and $k$, and a multiple$m_{n, k}$ also simple in its dependence on $n$. In addition, we prove thatboth equalities: $\mathrm{lcm}(n, n + 1, ..., n + k) = d_{n, k}$ and$\mathrm{lcm}(n, n + 1, ..., n + k) = m_{n, k}$ hold for an infinitely manypairs $(n, k)$.

Nontrivial lower bounds for the least common multiple of some finite sequences of integers