[seqfan] Re: Help needed... for proofs, Brocard's Conjecture, etc.

[M. F. Hasler]

On Tue, Dec 30, 2014 at 4:22 PM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> In https://oeis.org/A050216 "Number of primes between (prime(n))^2 and
> (prime(n+1))^2, with a(0) = 2 by convention." it is told that
> Brocard's Conjecture states that for n >= 2, a(n) >= 4.
>
> See also http://en.wikipedia.org/wiki/Brocard%27s_conjecture
>
> Now the question: for up to which k it is proved (or "obvious") that
> A050216(n) >= k, for all n >= 2 ?

I also don't understand Tony's comment
"
The lines in the graph correspond to prime gaps of 2, 4, 6, ... .  -
T. D. Noe, Feb 04 2008
"
Smaller gaps correspond to more primes and thus to higher lines. But
clearly there will be no interval [ prime(n)^2 ...prime(n+1)^2 ] with
only gaps of 2, since a gap of 2  (primes 6k+-1)  is followed by a gap
>= 4. So which line could correspond to gaps of 2, if not a "line"
consisting in only one point ?

Also, the larger the gaps, the less primes. but there seems one line
with minimal slope,
m(n) = min { A050216(k) ; k>n }
I can't see how this "line" could correspond to a given gap.
Numerically it seems that the lower line has a slope somewhere below
and/or close to m(n)/n ~ 1.8 for n large enough,
and m(n) > 1.7n for n>880, m(n) > 1.75n for n>2610.

Although I cannot exlude this, it does not look as if that slope would
grow up to 2.
Would this 2 be the value suggested by Tony ?
I.e., slopes (of the 1st, 2nd, 3rd... ray from below) that
asymptotically tend to 2, 4, 6, ... ?

The slopes of the different lines do indeed look as if they were
"equally spaced", i.e., multiples of the smallest one.
Yet, again, it seems not obvious why it would be these integer values,
and even less how they could correspond to given prime gaps.
I think it would be interesting to push further the study of these
lines and their slopes.
Also the relative density of these lines is interesting: the lowest
rays have the highest density,but e.g. the 9th, 12th and 15th line
seem to have higher density than their neighbors. It is known that
gaps of 30 and 210 are more frequent than others, but again the link
between these two lists is not obvious to me.

Maximilian