[seqfan] Re: All integers a(n) > 20 are in sequence A246778

[Vladimir Shevelev]

On the one hand, your result does more 
plausible the truth
of Firoozbakht's (1982)-conjecture. As you
proved, from this conjecture it follows that,
for all k>9,
p_(k+1) - p_k < log^2(p_k) - log (p_k) -1. (1)
On the other hand, A. Granvill (1995) in his 
improved Cramer's model obtained that 
lim sup (p_(k+1) - p_k)/log^2(p_k) >= 
2*e^(-c)=1.1229...,
which contradicts (1). Cf.
http://www.dartmouth.edu/~chance/chance
_news/for_chance_news/Riemann/cramer.pdf_
(Scand. Actuarial J., 1 (1995), 12-28).
Also, of course, Granvill's result cannot
contradict your last theorem for A246778.

Best regards,
 Vladimir


_______________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Alexei Kourbatov [akourbatov at gmail.com]
Sent: 28 November 2015 03:41
To: Sequence Fanatics Discussion list
Subject: [seqfan] All integers a(n) > 20 are in sequence A246778

Dear Seqfans,
Just added a new theorem in A246778 showing that all terms a(n)>20 are in
the sequence (just as conjectured by Farideh Firoozbakht) - provided that
prime gaps near x are less than x^0.75. Any feedback will be greatly
appreciated. (If you happen to have a nicer/shorter proof, that would be
great!)
Alexei

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