[seqfan] Re: A173897: Quasi-linear patterns in graph

[David Wilson]

Suppose the SG primes to have a slowly decreasing density d(n) around n (like the density 1/log(n) for primes).
a(n) counts the number of SG primes on the interval [p(n) ^2, p(n+1)^2], with p(n) = nth prime.
So we will have

a(n) 
=~ d(p(n)^2) * (p(n+1)^2 - p(n)^2)
= d(p(n)^2) * (p(n+1) + p(n)) * (p(n+1) - p(n))
=~ d(p(n)^2) * 2p(n) * (p(n+1) - p(n))

The first two factors change relatively smoothly, the last factor is the prime gap, and fluctuates more or less randomly between positive even integers.
So each of the kth "line" in the plot of a(n) likely includes the elements a(n) where the prime gap (p(n+1) - p(n)) = 2k.

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Zak
> Seidov via SeqFan
> Sent: Sunday, September 04, 2016 7:34 AM
> To: Sequence Fanatics Discussion list
> Cc: Zak Seidov
> Subject: [seqfan] A173897: Quasi-linear patterns in graph
> 
> Any idea about quasi-linear patterns in graph of A173897?
> --
> Zak  Seidov
> 
> --
> Seqfan Mailing list - http://list.seqfan.eu/