New sequence : Primes for which SQRT(A000040(n)) < A001223(n)

[reismann at free.fr]

Dear Dean and Seqfans

First of all, sorry for the multiple sendings, I had a problem with my e-mail.

Thank you for your answer. I will follow your advice and I like this definition
:
Primes p for which there are no primes between p and p+sqrt(p) :
3,7,13,23,31,113

About the Cramer's conjecture, it's the object of my comment on A117078 :
Primes p for which there are no primes between p and p+ln²(p) :
2,3,7
Primes p for which the decomposition in weight*level+gap
(A000040(n)=A117078(n)*A117563(n)+A001223(n)) is impossible :
2,3,7
I look for the relation between my decomposition and the Cramer's conjecture.

"You could add a comment that there are no other terms less than
218034721194214273.  (That's assuming that all of the terms in
http://www.research.att.com/~njas/sequences/b002386.txt  are correct.)"

srqt(p) > ln² p for p=2, p=3 and p in (5507, infinity)
sqrt(p) < ln²p for p in (5, 5503)
So by considering the Cramer's conjecture as true we have :
Primes p for which there are no primes between p and p+sqrt(p) < 5507.
Attached two graphs illustrating this comment.
Have a nice day.

Rémi EISMANN

Selon Dean Hickerson <dean at math.ucdavis.edu>:

> Mostly to Remi EISMANN:
>
> > I will submit this sequence in January :
> >
> > %I A000001
> > %S A000001 3,7,13,23,31,113
> > %N A000001 Primes for which SQRT(A000040(n)) < A001223(n).
> > %C A000001 Conjecture : this sequence is finite and complete.
> > %e A000001 a(1) = 3 because SQRT(3) < 2.
> > a(6) = 113 because SQRT(113) < 14.
> > %Y A000001 A000040(n), A001223(n).
> > %O A000001 1,1
> > %K A000001 ,fini,nonn,
> > %A A000001 Remi Eismann (reismann at free.fr), Dec 10 2006
> >
> > Any comments on the sequence or on the conjecture ?
>
> I suggest that you rewrite the definition so that people don't have to look
> up both A000040 and A001223 to figure out what it means.  Something like
> "Primes p for which sqrt(p) < q-p, where q is the smallest prime larger
> than p." or "Primes p for which there are no primes between p and
> p+sqrt(p).".
>
> You could add a comment that there are no other terms less than
> 218034721194214273.  (That's assuming that all of the terms in
> http://www.research.att.com/~njas/sequences/b002386.txt  are correct.)
>
> And you could state that finiteness of the sequence would follow from
> Cramer's conjecture that
>
>     lim sup (p(n+1)-p(n))/log(p(n))^2 = 1.
>
> Dean Hickerson
> dean at math.ucdavis.edu
>


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