This question is related to Is there a binary spigot algorithm for log(23) or log(89)? by Dan Brumleve.

A binary BBP-type formula is a convergent series formula of the type $$ \alpha=\sum_{k=0}^\infty\frac{p(k)}{2^kq(k)} $$ http://mathworld.wolfram.com/BBP-TypeFormula.html

The first prime for which it is not known whether such a formula exists is $23$. http://www.davidhbailey.com/dhbpapers/digits.pdf, http://oeis.org/A104885

A conditionally convergent formula for $log(n)$ is given by the generalized Mercator series

$$ log\left(n\right)=\sum_{k=0}^{\infty}\left(\sum_{i=nk+1}^{n(k+1)}\frac{1}{i}-\frac{1}{k+1}\right) $$ Do these series converge to logarithms?, Series for logarithms

For odd arguments such as $23$, partial sums are closer to the limit if the following regrouping is chosen

$$ log(2n+1)=H_n+\sum_{k=1}^{\infty}\left(\sum_{i=-n}^{n}\frac{1}{(2n+1)k+i}-\frac{1}{k}\right) $$


Setting $n=11$ above, $$ log(23)=H_{11}+\sum_{k=1}^{\infty}\left(\sum_{i=-11}^{11}\frac{1}{(2n+1)k+i}-\frac{1}{k}\right) $$

Series acceleration allows to obtain the binary BBP-type formula for $log(2)$ from Mercator series.

Q: How can a BBP-type formula for $log(23)$ be obtained through acceleration of the corresponding conditionally convergent series?


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