[seqfan] Density of de Polignac numbers

[Tomasz Ordowski]

Dear readers!

Wesolowski inequality in A254248
works only for some constant c <= 0.
For c = 0 we get density d < 1/e correctly.

I found a more sensible formula, namely.

The density d(n) of de Polignac numbers <= n
is d(n) < (1 - 2 / log n)^(log n / log 2). It works!
Their asymptotic density d(oo) < 0.055833...

Conjecture: d(n) ~ (1 - 2 / log n)^(log n / log 2).
The asymptotic density d(oo) = exp(-2 / log 2).
This density d = 0.055833... is hypothetical.

Any comments are welcome!

Best regards,

Thomas Ordowski
____________________
https://oeis.org/A254248
https://oeis.org/history/view?seq=A006285&v=73
https://math.dartmouth.edu/~carlp/coverbirthtalk.pdf