Lengths Of Runs In The #-Of-Divisors Sequence

[Joshua Zucker]

On 7/17/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
> On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:

> > It seems VERY likely to me that there is no infinite string of 1's, or of
> > anything else, in sequence A131789 (ie. the terms of A131790 are all
> > finite).
> >
> > Can it be PROVED that all terms of A131790 are finite, possibly using
> > Hardy and Wright or some other such reference?
>
> No need for Hardy and Wright, just Euclid -- there are infinitely many
> primes.  Plus there are not any consecutive primes after 2.  Hence the
> primes (2s) partition the list into finite chunks.

Wups, I think that's a proof for A131789, not for A131790.

--Joshua



Hello all,

I was looking at A069803 - Smaller of two consecutive palindromic primes:	2, 3, 5, 7, 181, 787, 919 
Conjectured to be complete. 

I am interested in seeing a proof that 919 is actually the largest palindromic prime such that the next prime is palindromic. 
I checked up to 10^8 with Mathematica coding. 

Also, it is obvious that the distance from a palindrome n to the next one is more than Sqrt(n/10). It is clear that prime gaps grow slower than that. Looking at the prime gaps sequence A053303, it is easy to prove that 919 is the last number like that up to 10^16.

Is there a bound for prime gaps that proves that the gaps are less than Sqrt(n/10) starting from some n?

Tanya


_________________________________________________________________
Need personalized email and website? Look no further. It's easy
with Doteasy $0 Web Hosting! Learn more at www.doteasy.com