There are exactly four 2-term Machin-like formulas where the arctan argument is a unit fraction, namely, $$\begin{aligned} \frac{\pi}{4} &=\arctan\frac{1}{2} + \arctan\frac{1}{3}\\ &=2 \arctan\frac{1}{2} - \arctan\frac{1}{7}\\ &=2 \arctan\frac{1}{3} + \arctan\frac{1}{7}\\ &=4 \arctan\frac{1}{5} - \arctan\frac{1}{239} \end{aligned}$$
(Updated question:) Using a rather basic Mathematica code, I found twenty-four 3-term formulas using only unit fractions and the constraint that one denominator $d>100$. However, almost half use $d=239$, namely
$$\begin{aligned} \frac{\pi}{4} &=4 \arctan\frac13 -4 \arctan\frac18-\arctan\frac1{239}\\ &=2\arctan\frac17+4\arctan\frac18+\arctan\frac1{239}\\ &=4\arctan\frac17+4\arctan\frac1{18}-\arctan\frac1{239}\\ &=8\arctan\frac18-4\arctan\frac1{18}+3\arctan\frac1{239}\\ &=4\arctan\frac14-4\arctan\frac1{21}-\arctan\frac1{239}\\ &=4\arctan\frac16+4\arctan\frac1{31}-\arctan\frac1{239}\\ &=6\arctan\frac17-4\arctan\frac1{57}+\arctan\frac1{239}\\ &=6\arctan\frac18+2\arctan\frac1{57}+\arctan\frac1{239}\\ &=12\arctan\frac1{18}+8\arctan\frac1{57}-5\arctan\frac1{239}\\ &=8\arctan\frac1{10}-4\arctan\frac1{515}-\arctan\frac1{239}\\ \end{aligned}$$
with the third to the last appearing here, the second to the last being Gauss's Machin-Like formula, and the last in here (courtesy of Leibovici's comment).
Questions:
- Is the list above complete for $d=239$?
- And why do so many formulas use $d=239$?