# [seqfan] Re: Is there a phenomenon?

Mon Sep 19 11:09:02 CEST 2011

```I don't understand your last sequence " Primes having an  r>1.” For large prime p,  the next prime q and a number r>1, the interval (r*p,r*q) in average has length r*ln(p). Therefore, if  not to fix r, then every

prime is a potential candidate to your sequence for a some value of r. In my conjecture (in comment to A195325) a key word is "least". I noted that the number of the first r-gap primes for r=2,3,...which are lesser's of twin primes on average grows with r
and posed this conjecture. After Charles’s check up to r=10^10 there is a more certainty that it is true.

Best regards,

----- Original Message -----
From: RGWv <rgwv at rgwv.com>
Date: Sunday, September 18, 2011 23:55
Subject: Re: [seqfan] Is there a phenomenon?
To: Vladimir Shevelev <shevelev at bgu.ac.il>

>
>     A bigger question is, “What Primes have an
> r>1?”
>
>     The sequence begins:  29, 59, 71, 101,
> 107, 137, 149, 179, 191, 197, 227, 239, 263, 269, 281, 311, 347,
> 379, 419, 431, 443, 461, 499, 521, 557, 569, 599, 617, 641, 659,
> 673, 727, 757, 809, 821, 823, 827, 853, 857, 881, 883, 907, 967,
> 977, 991, 1009, 1019, 1031, 1049, 1061, 1091, 1093, 1151, 1213,
> 1229, ..., .
>
>     Of a greater interest, this sequence does not
> include just the lesser of the twin primes. Therefore Charles’s
> conjecture is probably wrong.
>
> Bob.

----- Original Message -----
From: zak seidov <zakseidov at yahoo.com>
Date: Sunday, September 18, 2011 8:16
Subject: [seqfan] Re: Is there a phenomenon?
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> "conjecture that always a_3(n)<=a_2.5(n)" is wrong.
> Zak
>
> ----- Original Message -----
> > From: Vladimir Shevelev <shevelev at bgu.ac.il>
> > To: seqfan at list.seqfan.eu
> > Cc:
> > Sent: Saturday, September 17, 2011 11:59 PM
> > Subject: [seqfan] Is there a phenomenon?
> >
> > For a real r>1, let us call a prime p an r-gap prime, if there
> is no prime
> > between r*p and r*q, where q is the next prime after p. A few
> the first terms of
> > 3-gap primes are
> > 71, 107, 137, 281, 347, 379, 443, 461, 557, 617, 641, 727,
> 809, 827, 853,...,
> > while the first terms of 2.5-gap primes are
> > 127, 197, 281, 311, 347, 431, 613, 659, 673, 739, 877, 991,
> 1049, 1229, 1277.
> > Looking at these tables I did a conjecture that always
> a_3(n)<=a_2.5(n),
> > where the equality holds only for n=5. But, in cases when
> a_3(n)<5/6*a_2.5(n)
> > this means that the longer prime gaps of a certain form appear
> earlier than the
> > shorter ones. Thus, if the inequality a_3(n)<5/6*a_2.5(n)
> occur more often
> > than the opposite one, then we have a phenomenon.
> > Can anyone verify such inequality for larger n and, maybe,
> disprove this
> > phenomenon?
> >
> > Best regards,
> >
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
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