[seqfan] Re: Primes and prime remainders

[M. F. Hasler]

Bob,
Vladimir noticed that El Bachraoui's theorem of existence of a prime p
in [2n,3n] does not imply that a(n+1)=2a(n)+p with p<a(n), because
here the additional constraint of p-2a(n) = prime applies.
PS: See https://oeis.org/A248370 for the sequence.

Maximilian

On Sun, Oct 5, 2014 at 6:26 PM, Bob Selcoe <rselcoe at entouchonline.net> wrote:
> Hi Vladimir, Eric, et. al,
>
>> But is there,
>> for prime k>=5, such a prime q such that q-2*k is a prime (r)?
>> ... So, till now it is an unsolved problem.
>
>
> Vladimir, what are you looking to solve?  How does this pertain to Eric's
> sequence?
>
> The number of ways q can be written as  q-2*k = r (k,q,r are prime) is
> A103274.  Are you looking for an equation for A103274??   How does this
> pertain to Eric's sequence?
>
> Cheers,
> Bob
>
> --------------------------------------------------
> From: "Vladimir Shevelev" <shevelev at bgu.ac.il>
> Sent: Sunday, October 05, 2014 1:47 PM
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Primes and prime remainders
>
>> If a(n) is lesser of twin primes, then a(n+1)=a(n)+2;
>> otherwise, it is intuitively clear that a(n+1) is
>> the smallest prime of the form 2*a(n)+prime.
>> Indeed,it is known that between 2*k and 3*k, k>=2,
>> there is a prime (q )( El Bachraoui (2006)). But is there,
>> for prime k>=5, such a prime q such that q-2*k is a prime (r)?
>> In this case we have q=r+2*k, where q,k,r are primes.
>> It is similar to a special case of the  Lemoine's-Levy's
>> conjecture for odd prime (here q is prime)
>> [cf. A046927], but with additional condition r<k.
>> So, till now it is an unsolved problem.
>>
>> Regards,
>> Vladimir
>>
>> ________________________________________
>> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Eric Angelini
>> [Eric.Angelini at kntv.be]
>> Sent: 05 October 2014 12:03
>> To: Sequence Discussion list
>> Subject: [seqfan] Primes and prime remainders
>>
>> Hello SeqFans,
>> Primes p(n) such that the remainder
>> of p(n)/p(n-1) is prime.
>> The seq P starts with p(1)=3 and is always
>> extended with the smallest possible
>> prime.
>>
>> P=3,5,7,17,19,41,43,89,...
>>
>> Example:
>> 5/3--> remainder 2
>> 7/5--> remainder 2
>> 17/7--> remainder 3
>> 19/17--> remainder 2
>> 41/19--> remainder 3
>> 43/41--> remainder 2
>> 89/43--> remainder 3
>> etc.
>> Hope I didn't mistake somewhere.
>> Best.
>> É.