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

[Robert Israel]

But we do know (or can compute, e.g. in some way that doesn't 
mention the word "prime") which numbers are prime, so this
clarification doesn't mean anything.

Robert Israel                                israel at math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada

On Tue, 11 Dec 2007, Andrew Plewe wrote:

> That's technically correct but it presupposes knowledge of which numbers are
> prime in order to generate the members of the sequence. Thus a
> clarification; generating the sequence shouldn't require that we know which
> numbers are prime.
>
> 	-Andrew Plewe-
>
> -----Original Message-----
> From: Robert Israel [mailto:israel at math.ubc.ca]
> Sent: Tuesday, December 11, 2007 12:08 PM
> To: Jonathan Post
> Cc: Andrew Plewe; seqfan at ext.jussieu.fr; jonathan post
> Subject: Re: Most "compact" sequence such that there is at least one prime
> between a(n) and a(n+1)
>
> What's the prime between 8 and 9 then?
> But how about a(n) = p(n)-1 where p(n) is the n'th odd prime?
>
> Robert Israel                                israel at math.ubc.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel
> University of British Columbia            Vancouver, BC, Canada
>
> On Tue, 11 Dec 2007, Jonathan Post wrote:
>
>> By definition, your answer would seem to be:
>>
>> A018252  The nonprime numbers (1 together with the composite numbers
>> of A002808).
>> 	1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27,
>> 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50,
>> 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72,
>> 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88
>>
>> On 12/11/07, Andrew Plewe <aplewe at sbcglobal.net> wrote:
>>> "Compact" here means the smallest difference between terms. One candidate
> is
>>> A052548: 3, 4, 6, 10, 18, etc. It is easy to show that there is one prime
>>> between successive members of this sequence by application of Bertrand's
>>> Postulate (at least one prime between n and 2n - 2, n >=3):
>>>
>>> 3, 3*2 - 2 = 4
>>> 4, 4*2 - 2 = 6
>>> 6, 6*2 - 2 = 10
>>>
>>> etc.
>>>
>>> Are there any sequences more "compact" than this that can be proved to
> have
>>> primes between successive members?
>>>
>>>
>>>         -Andrew Plewe-
>>>
>>>
>>>
>>>
>>
>
>