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  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.
 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)