Is there a simple formula for $\mathsf{lcm}(1,2,\dots,n-1,n)$ that can be explicitly stated out?
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1Related : http://math.stackexchange.com/questions/834220/least-common-multiple-lim-sqrtn1-2-dotsc-n-e – Watson Feb 19 '16 at 09:56
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See OEIS A$3418$. – Lucian Feb 19 '16 at 12:39
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This is Landau's function. It can be written as $$ lcm(1,2,\ldots,n) = \prod_{p\le n} p^{\lfloor \log_p n \rfloor} $$ and so $$ lcm(1,2,\ldots,n) = e^{\psi(n)} $$
where $\psi$ is the second Chebyshev function: $$ \psi(x) = \sum_{p^k\le x}\log p = \sum_{p\le x}\lfloor\log_p x\rfloor\log p, $$
There is no explicit formula for $\psi$, but there are asymptotic results. For instance, $$ \psi(x) \sim x $$ which is equivalent to the prime number theorem. More detailed asymptotics are related to the Riemann Hypothesis.
lhf
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maybe writing $\displaystyle e^{\psi(n+\epsilon)}$ would be more careful – reuns Feb 19 '16 at 12:30
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@ lhf : because in nearly every formula it is $\psi_0(x)$ which appears so it is safer to write $\psi(n+\epsilon)$ – reuns Feb 19 '16 at 12:38