# [seqfan] Re: 5, 20, 120, 540, 6480, ...

M. F. Hasler oeis at hasler.fr
Wed Mar 30 22:46:25 CEST 2022

```I just noticed that :
X = { n  such that  p(n+1)+p(n) is not divisible by p(n+1) - p(n) }
= { 4, 6, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 23, 24, 25, 27, 29,
30, 31, 32, ... }
=?= A049579 : Prime subscripts for which residue of (prime(n)-1)!+1
modulo prime(n)+2 equals 1.
(equality is only apparent, but not obviously / proven to be true:
I don't see immediately why this is the same, because the prime(n+1) does
not at all enter the definition of a(n) in the second case...)

Also notice that according to the first terms of X, it is rather rare that
( p(n+1)+p(n) ) /  ( p(n+1) - p(n) )
is non-integer, but later this is almost always the case! For that reason,
I'm not so sure that it couldn't happen that a prime gap of the form
2p, with some medium-sized prime, could occur before, or more often, than
the factor p in p(n)+p(n+1), at a certain point.

-Maximilian

On Wed, Mar 30, 2022 at 1:29 PM Robert Gerbicz <robert.gerbicz at gmail.com>
wrote:

> Hi !
>
> See:
>
> https://terrytao.wordpress.com/2016/03/14/biases-between-consecutive-primes/
> and from that page: https://arxiv.org/pdf/1603.03720.pdf conjecture 1.1
> what they have in conjecture 1.1 is that for consecutive p1,p2 primes you
> will see p2+p1 is divisible by q more often than p2-q1.
>  [because in the latter case p2==p1==a mod q, while in the other case p2
> and p1 are in different residue classes if q>2].
> You'd still need effective constants on that conjecture's bounds, but at
> least we see why this should be true,
> notice also that p(n+1)-p(n) is "small", so you could prove that the
> product is an integer up to a pretty large bound, just factorize the terms
> using prime up to L, if p(n+1)-p(n)<=L is true for all n<=N.
>
>
> Tomasz Ordowski <tomaszordowski at gmail.com> ezt írta (időpont: 2022. márc.
> 30., Sze, 17:07):
>
> >
> > Let a(n) = Product_{k=1..n} (prime(k+1)+prime(k))/(prime(k+1)-prime(k)).
> > Conjecture: a(n) is an integer for every natural n.
> > Is it known or provable?
> >
> > Best regards,
> >
> > Thomas Ordowski
> >
> >