Which Ramanujan's formula used the biggest constants?

Question

After the first time I saw the movie "The Man Who Knew Infinity", about Srinivasa Ramanujan, I've looked up some of his formulas on the web. One of such formulas amazed me the most, because it used three really big integers (like 10 to 20 digits each), called A, B and C and indicated below the formula. I can't remember if it was a formula for pi, partitions or what, but I think it added some 50 digits each iteration. I just can't find this formula anymore. Can anyone tell me which of Ramanujan's formulas used the biggest constant integers in it?

Answer 1

I can not be certain but you are probably remembering something like a formula in MathWorld Pi Formulas. Near the end of that article is equations (93) to (96). The first equation is

$$ \frac{\sqrt{-C^3}}\pi = \sum_{n=0}^\infty \frac{(6n)!}{(3n)!(n!)^3} \frac{A+nB}{C^{3n}}, \tag{93}$$ The next three equations have numbers with up to 50 decimal digits. Immediately after is this

This gives 50 digits per term. Borwein and Borwein (1993) have developed a general algorithm for generating such series for arbitrary class number.

Consult the MathWorld link for the actual values of the three numbers $A,B,C$.

Answer 2

One of the formulas of such nature is \begin{align} 7709321041217+32640\Phi_{0, 31}(x) &= 764412173217Q^{8}(x)\notag\\ &\,\,\,\,\,\,\,\,+ 5323905468000Q^{5}(x)R^{2}(x)\notag\\ &\,\,\,\,\,\,\,\,+ 1621003400000Q^{2}(x)R^{4}(x)\notag \end{align} where \begin{align} P(x) &=1-24\sum_{n=1}^{\infty}\frac{nx^n}{1-x^n}\notag\\ Q(x) &=1+240\sum_{n=1}^{\infty} \frac {n^3x^n}{1-x^n}\notag\\ R(x) &=1-504\sum_{n=1}^{\infty} \frac {n^5x^n}{1-x^n}\notag\\ \Phi_{0,r}(x)&=\sum_{n=1}^{\infty}\frac {n^rx^n} {1-x^n}\notag \end{align}