What about irrationality of $a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$?

Question

This question is related to my question here which depend on the convergence of this sequence:$a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$ however the limit of convergence is not clear even now, But my question here is to ask about irrationality of that sequence ? its clear to me that is rational only for $ n=1$ ,What about $ n >1$ ?

Answer 1

Clearly rational only for n=1?

This is more a check on what is known about these numbers rather than an answer. We have:-

$n=1$ Rational

$n=2$ Irrational, algebraic

$n=3$ Transcendental

$n=4$ ????

So it is clearly irrational for $n=2$ and $3$. However, have you made any progress on the case $n=4$?