Machin's formula (M000001) is a formula for π that was discovered by John Machin in 1706: $$\pi=4\arctan\left(\frac15\right)-16\arctan\left(\frac1{239}\right)$$ There are a huge number of Machin-like formulae that have since been discovered, all of which are of the form: $$\pi=a_0\arctan(b_0)+a_1\arctan(b_1)+a_2\arctan(b_2)+\dots$$ where the \(a\)s and \(b\)s are rational numbers.
On the page for each Machin-like formula, you will find the formula in compact notation. In this notation, a[b] represents \(a\arctan(1/b)\). For example, the compact formula 4[5] - 16[239] represents Machin's formula, ie $$\pi=4\arctan\left(\frac15\right)-16\arctan\left(\frac1{239}\right)$$
Lehmer's measure as introduced in 1938[1] as a measure of how computationally efficient a Machin-like formula is, with lower values corresponding to more efficient formulae.
For the general Machin-like formula $$\pi=a_0\arctan(b_0)+a_1\arctan(b_1)+a_2\arctan(b_2)+\dots$$ Lehmer's measure is computed using $$\sum_i\frac1{\log_{10}\left(\frac1{b_1}\right)}.$$ For example, for Machin's formula, Lehmer's measure is $$\frac1{\log_{10}(5)}+\frac1{\log_{10}(239)}=1.85112$$