[seqfan] Re: A play of twin primes?

[Maximilian Hasler]

The sequence of values in steps of 1/100, e.g.,

> F(1.99)=2104, F(2.00)=2160, F(2.01)=2079;

does not represent the evolution of the function:
You have, e.g., F(2 + 1/8000) = 2207
which "contradicts" the hypothesis that F has a local max at 2.00.

Maximilian


On Sat, Aug 6, 2011 at 1:06 PM, Vladimir Shevelev <shevelev at bgu.ac.il> wrote:
> Dear SeqFuns,
> I would like to tell you about an astonishing experiment.
> For a real  number u>1, I call a prime p a u-gap prime, if, for the next prime q>p, the interval (u*p,u*q) contains no primes. I noticed some abnormality in  the distribution of u-gap primes with respect to the values of u. Consider, say, the first 10^5 primes (in order to speak about statistics). Denote F(u)=sum_{n<=10^5: p_n is u-gap}1. Let u change with step 0.01. Then in the neighborhoods of integers we have
> F(1.99)=2104, F(2.00)=2160, F(2.01)=2079;
> F(2.99)=1302, F(3.00)=1539, F(3.01)=1346;
> F(3.99)=  926, F(4.00)=1033, F(4.01)=  918.
> On the other hand,  in the neighborhoods of half-integers we have
> F(1.49)=2796, F(1.50)=2703, F(1.51)=2713;
> F(2.49)=1669, F(2.50)=1580, F(2.51)=1657;
> F(3.49)=1149, F(3.50)=1101, F(3.51)=1131;
> F(4.49)=  802, F(4.50)=  786, F(4.51)= 809.
> Thus we observe local maximums in integers and local minimums in half-integers!
> If anyone to confirm these observations and to try to explain these phenomenons?
> I think that it is a play of twin primes, but I have no a quite satisfactory explanation.
>
> Regards,
> Vladimir Shevelev
>
>  Shevelev Vladimir‎
>
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PS: For my records (by default this F ignores the untypical primes 2,3,5)
F(u,LIM=10^5,p=5,c=0)=forprime(q=nextprime(p+1),LIM,nextprime(u*p)>u*q&c++;p=q);c