TITLE: Prime numbers making constant : 1.2527
QUESTION [6 upvotes]: Reading "Excursion in calculus" (Robert M. Young, 1992), exercice 13 on page 71 ask the reader to show there is a constant $c\approx 1.25$ such that

$a_0=2^c$
$a_{n+1}=2^{a_n}$
$\forall n\; \lfloor a_n\rfloor$ is prime

This can be proved using Bertrand's postulate.
I found that $c\approx 1.2527$ is a good candidate as

$\lfloor a_0\rfloor=2$
$\lfloor a_1\rfloor=5$
$\lfloor a_2\rfloor=37$
$\lfloor a_3\rfloor=153703792531$ that is prime too.

There must be several different constants that have this property, but we can suppose that $c$ is minimal.
Does that constant has a name ? Does someone computed or is trying to compute more decimal ?

REPLY [5 votes]: This number is called Bertrand's number in OEIS; the decimal expansion is OEIS A$079614$ and is
$$1.251647597790463017594432053623346969\ldots\;.$$
The sequence of associated primes is OEIS A$051501$. There are references in both OEIS entries.