Estimating Fouvry's constant

[Pierre Abbat]

See A073024 (Fouvry primes) and A387952 (Cài semiprimes).

I've been reading Cài's paper and trying to understand it, and I wonder how
common Fouvry primes are among the primes. I haven't read Fouvry's paper, but
from what A073024 and Cài said about it, Fouvry simply proved that the
abundance is positive. So I added another function to the Julia program in
A387952 and ran it with arguments up to 2^30, and Fouvry's constant seems to
be somewhere between 0.372 and 0.38. Can anyone figure out a faster way to
compute the constant than listing all the primes in an interval and counting
how many are Fouvry primes?

Pierre
--
li ze te'a ci vu'u ci bi'e te'a mu du
li ci su'i ze te'a mu bi'e vu'u ci

[sven-h...@gmx.de]

Hello,
searching with 'Fouvry constant' in Google shows a discussion in MathOverFlow which mentions A073024, linked here:

https://mathoverflow.net/questions/458013/density-of-primes-p-where-p-1-has-a-prime-factor-exceeding-p2-3

with a table of values and an answer with more details and an approximation.
Sven

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[Hugo Pfoertner]

The table in the mathoverflow discussion seemed a bit short to me. Therefore, I created https://oeis.org/A395819 . It has a link to a version of Fouvry's classic article that is not behind a paywall.  If you naively extrapolate the ratios using the default settings of Mathematica's sequence limit function, you end up surprisingly close to the conjectured asymptotic limit of log(3/2) (A016578) derived in the mathoverflow discussion.

ResourceFunction["SequenceLimit"] [{0.00000, 0.20000, 0.28571, 0.31164,
   0.33559, 0.34968, 0.35903, 0.36574, 0.37076, 0.37460, 0.37766,
  0.38018}, Method -> Automatic]

0.404938