User blog comment:TechKon/Lambda Function/@comment-25418284-20140227233334

At the beginning you write that your function "defines how many observable universes the digits of a number can fill, if the digits were all the size of a Planck length, the smallest physically possible length." This implies the function \(n \mapsto \lfloor \log_10 n + 1 \rfloor / \left(\frac{4}{3} \pi R^3 / \ll_P^3 \right)\), where \(R\) is the radius of the OU and \(\ll_P\) is the Planck length. This is a weak slow-growing function by googological standards, reaching a logarithmic growth rate.

But at the end you reverse this definition. Now \(n \Lambda(m)\) is "the number of \(n\)'s that can fill \(m\) observable universes," which is entirely different. This gives the definition \(n \Lambda(m) = m \left(\frac{4}{3} \pi R^3 / \ll_P^3 \right) / \lfloor \log_10 n + 1 \rfloor\).

Can you explain more formally how your function works?