A recent question asked for the Taylor expansion of $$f(x)=\exp\left(-\pi\,\frac {_2F_1(\frac12, \frac12, 1; 1-x)}{_2F_1(\frac12, \frac12;1;x)}\right)=\exp\left(-\pi\,\frac{K(1-x)}{K(x)}\right)=\frac{x}{16} \sum_{n=0}^\infty \frac {a_n} {2^{b_n}}\,x^n$$ The numerators $a_n$ correspond to sequence $A002639$ in $OEIS$. According to Jean-François Alcover's comment, $f(x)$ is the Taylor expansion of the $EllipticNomeQ(x)$ function in Mathematica (it gives the nome $q$ corresponding to the parameter $x$ in an elliptic function. In fact, the first terms of $f(x)$ are given here.
The $b_{2n}$ make the sequence $$\color{blue}{\{0,6,15,21,32,38,47,53,64,70,79,85,93,99,109,115,128,134,143,149,159,165,174,180,192 ,198\}}$$ and $b_{2n+1}=b_{2n}+1$.
The only thing I noticed is a pattern for $c_n=8n- b_{2n}$ $$\{0,2,1,3,0,2,1,3,0,2,1,3,3,5,3,5,0,2,1,3,1,3,2,4,0,2\}$$
My question is : what are the $b_n$ corresponding to ?