# [seqfan] Re: Primes and prime remainders

Bob Selcoe rselcoe at entouchonline.net
Sun Oct 5 23:36:07 CEST 2014

> 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.

Thus twin primes are remainder r=2, else non-twins.  Smallest r for
non-twins is 3.  Candidate numbers in P of this form are A023204  - let's
call them "fraternal twins".  ;-)

Extending sequence P to a(n)=6*10^9 (done so by hand; hopefully no
calculation errors):

P = 3, 5, 7, 17, 19, 41, 43, 89, 181, 367, 739, 1481, 1483, 2969, 2971,
5953, 11909, 23831, 23833, 47713, 95429, 190871, 381749, 763559, 1527121,
3054283, 6108607, 12217327, 24434701, 48869413, 97738843, 195477691,
390955399, 781910809, 1563821621, 3127643381, 3127643383

Sequence R (of remainders r) =
2,2,3,2,3,2,3,3,5,5,3,2,3,2,11,3,13,2,47,3,13,7,61,3,41,41,113,47,11,17,5,17,11,3,47,139,2

Nothing like sequences P and R appear in the OEIS.  Perhaps both could be

Might there be something worth exploring about fraternal primes in P?

Cheers,
Bob Selcoe

--------------------------------------------------
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,
>
> ________________________________________
> 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.
>
>
> Catapulté de mon aPhone
>
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