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

[Tomasz Ordowski]

P.S. See my draft
A352743 - OEIS <https://oeis.org/draft/A352743>
Thomas

czw., 31 mar 2022 o 13:05 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

>
> Primes p such that p+2 divides (p-1)!
> are primes p such that q-p does not divide q+p,
> where q is the next prime after p.
> These are the primes p for which q-p > 2.
> Cf. A049579 - OEIS <https://oeis.org/A049579> (see my draft).
>
> Thomas Ordowski
>
> czw., 31 mar 2022 o 05:26 Allan Wechsler <acwacw at gmail.com> napisał(a):
>
>> The observation about the smallness of p(n+1) - p(n) is very convincing to
>> me, and turns me into a believer in this conjecture, but it feels like it
>> would be very hard to prove, like the twin prime conjecture (on which Tao
>> and colleagues have also made progress) and the Goldbach conjecture (about
>> which I have heard no news for a very long time).
>>
>> This makes me suspect that similar conjectures will be true for other
>> just-slightly-supralinear sequences.
>>
>> 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):
>> >
>> > > Dear readers!
>> > >
>> > > 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
>> > >
>> > >
>> > > <#m_-7681193058414376069_DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2>
>> > >
>> > > --
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>> > >
>> >
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>>
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>