Here’s a surprising result: The least common multiple of the first *n* positive integers is approximately exp(*n*).

More precisely, let φ(*n*) equal the log of the least common multiple of the numbers 1, 2, …, *n*. There are theorems that give upper and lower bounds on how far φ(*n*) can be from *n*. We won’t prove or even state these bounds here. See [1] for that. Instead, we’ll show empirically that φ(*n*) is approximately *n*.

Here’s some Python code to plot φ(*n*) over *n*. The ratio jumps up sharply after the first few values of *n*. In the plot below, we chop off the first 20 values of *n*.

from scipy import arange, empty from sympy.core.numbers import ilcm from sympy import log import matplotlib.pyplot as plt N = 5000 x = arange(N) phi = empty(N) M = 1 for n in range(1, N): M = ilcm(n, M) phi[n] = log(M) a = 20 plt.plot(x[a:], phi[a:]/x[a:]) plt.xlabel("$n$") plt.ylabel("$\phi(n) / n$") plt.show()

Here’s the graph this produces.

[1] J. Barkley Rosser and Lowell Schoenfeld. Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics, Volume 6, Issue 1 (1962), 64-94. (On Project Euclid)

Interesting to compare this with Stirling’s formula, as a measure of the asymptotic badness of $n!$ as a first guess or upper bound on $\text{lcm}(1:n)$.

Isn’t \phi(n) a bad choice of notation here, given Euler’s totient function?

Amazing tidbit of knowledge! (sorry for the content-less remark)

Very nice, indeed. I did not know that. But I am far from an expert in this kind of number theory.

For n=1000000000 we have phi(n)/n = 0.9999824279662022 (obviously by a method different from the method in the post).

@David Malone: Yes, it’s unfortunate notation, but it matches the paper I referenced.

In the paper you reference, phi(n) denotes the number of positive integers <= n and relatively prime to n.

In the paper, psi(n) is the logarithm of the least common multiple of all positive integers <= n.

It seems you are using the inequality

(1-epsilon)x< psi(x) e^5000.

Do you know if it tends to a limit?