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one of the most prominent functions of the first $n$ natural numbers is the factorial $n!$ that denotes their product.

Today however I wondered whether the least common multiple $\mathrm{lcm}(n):=\mathrm{lcm}(\{i\in\mathbb{N}\,|\,1\leqq i\leqq n\})$ of the first $n$ natural numbers has already been the subject of mathematical studies.
The first elements of their sequence are $1,\, 2,\, 6,\,12,\,60,\,60,\,420,\,840,\,2520,\,2520,\,27720,\, \dots$ and it is A003418 – in the OEIS, as Christian Bernert pointed out in his comment.

Questions:

  • is there already an established name for $\mathrm{lcm}(n)$?
  • is there an agreed upon notation?
  • does it have an extension for non-integral complex values, analogous to the gamma function?
  • what can be said about bounds on, resp. growth-rate of the number of consecutive equal values apart from their relation to prime-gaps?

It is not hard to see that $\mathrm{lcm}(n)$ equals the procuct of maximal prime powers less or equal n

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    $\begingroup$ This is a long- and well-studied sequence and I could not believe it is not yet in OEIS. And of course it is: oeis.org/A003418 $\endgroup$
    – Christian Bernert
    Commented Aug 29, 2020 at 14:15
  • $\begingroup$ (Also the comments there should answer some of your questions.) $\endgroup$
    – Christian Bernert
    Commented Aug 29, 2020 at 14:18
  • $\begingroup$ @ChristianBernert maybe because entered it differently; I will edit my question accordingly. $\endgroup$
    – Manfred Weis
    Commented Aug 29, 2020 at 14:20
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    $\begingroup$ Just a formatting thing, you already have bold Questions and bullet points to add emphasis, but the addition of blockquotes sort of takes away emphasis. I know people used blockquotes before add emphasis because it put everything in a big yellow box but now it just grays the text out and moves it over a bit. $\endgroup$
    – user35370
    Commented Aug 29, 2020 at 14:48
  • $\begingroup$ This is similar to primorial. $\endgroup$
    – Emil Jeřábek
    Commented Aug 29, 2020 at 16:03

1 Answer 1

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The quantity $\operatorname{lcm}(n)$ is equal to $e^{\psi (n)}$, where $\psi$ is the second Chebyshev function. This function is well studied, and the prime number theorem is equivalent (indeed, usually proved using this equivalence) to the fact that $\psi (x)\sim x$.

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