# [seqfan] Re: Is there a phenomenon?

Sun Sep 18 22:36:48 CEST 2011

```Thank you, Zak!

It follows from the context that I wonder if there exist those values of n, for which among the first n numbers i the inequalities of the form a_3(i)>a_2.5(i) are in the majority (and the same for the inequalities a_3(i)>5/6a_2.5(i)).   It is interesting also a sequence of points n in which the difference a_2.5(n)-a_3(n)  (difference 5/6a_2.5(n)-a_3(n)) changes the sign.

Regards,

----- 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,
> >
> >
> >
> > _______________________________________________
> >
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> >
>
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>

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