[seqfan] Density of Fortunate primes

[Charles Greathouse]

The Fortunate primes,
https://oeis.org/A046066 ,
seem to have an asymptotic density, probably close to 1/2.  I was
interested in what could be proven, conditionally or otherwise.

As far as I know the sequence is not even known to be infinite.  Under
Fortune's conjecture (that all terms of A005235 are prime) the
sequence is at least known to be infinite, since prime(n) < A005235(n)
< prime(n)#.  That second inequality comes from Bertrand's postulate,
but this (or even Baker-Harman-Pintz) gives very weak bounds on
A046066.  As far as I know it's possible that a long string of terms
in A005235 are equal to some k, until term pi(k), at which point all
terms are k1 for a long string, where k1 is the longest prime gap
allowed by the relevant bound.  This leads to a tetrational bound in
A046066: something like k, exp(0.525 k), exp(.525 exp(0.525 k)), ....
Even the Riemann hypothesis only serves to decrease the base from
exp(0.525) = 1.69... to exp(0.5) = 1.64....

Surely more is possible.  At one point I convinced myself that a
double-exponential bound was possible under Fortune's conjecture.  Was
that achievable, or is this another proof that didn't fit in the
margin?

Charles Greathouse
Analyst/Programmer
Case Western Reserve University