I do not know if the problem has been posed or solved before, but I would like to launch a Polymath project about proving that
$$ a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}} $$
is a convergent series which it is assigned the two following sequences in OEIS [odd sequences][1] and [even sequence][2]. A quite detailed description of some attempts can be found [on MSE][3].


[3]:https://math.stackexchange.com/q/2822112/156150
[1]:https://oeis.org/A328942
[2]:https://oeis.org/A328941