Most "compact" sequence such that there is at least one prime between a(n) and a(n+1)

[Andrew Plewe]

sequence which is provably the "most" compact. Rather, I'm curious about the
	-Andrew Plewe-
From: Maximilian Hasler [mailto:maximilian.hasler at gmail.com]
Sent: Wednesday, December 12, 2007 6:21 AM
To: Hugo Pfoertner
Cc: seqfan
Subject: Re: Most "compact" sequence such that there is at least one prime
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From: "David W. Wilson" <wilson.d at anseri.com>
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Subject: RE: Most "compact" sequence such that there is at least one prime between a(n) and a(n+1)
Date: Wed, 12 Dec 2007 14:52:13 -0500
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My apologies.

The problem is to find exponent E such that interval (n^E, (n+1)^E] includes a prime for every n >= 1. The algorithm finds.

I wrote a lazy binary search which found E = log(541)/log(52), but this value appears to be incorrect.

I wrote a more methodical algorithm to find E, which finds

    E = log(1151)/log(95) = 1.5477771087+

The algorithm shows that no exponent e < E is admissible. Also, if we compare the size of the interval (n^E, (n+1)^E] to size of prime gaps listed in A005250 and A002386, we find it is much larger, indicating that we will find primes on (n^E, (n+1)^E] up to n^E = 2*10^16. Also, the prime gaps seem to be growing more slowly than (n+1)^E-n^E, so we can have a very warm fuzzy that E is the correct value.

Maybe someone could verify my findings.

Also, the b-files for A005250 and A002386 need to be verified against one another. They should be of the same length, and I'm not sure the primes and gaps listed in them match up.

> -----Original Message-----
> From: David W. Wilson [mailto:wilson.d at anseri.com]
> Sent: Wednesday, December 12, 2007 9:36 AM
> To: seqfan at ext.jussieu.fr
> Subject: RE: Most "compact" sequence such that there is at least one
> prime between a(n) and a(n+1)
> 
> Oh, wait, I meant to say...
> 
> Empirically, log(541)/log(52) = 1.59267+ seems to be a lower bound on
> exponents that admit at least one prime between n^e and (n+1)^e for all
> n >= 1.