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

[Andrew Plewe]

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-
>>
>>
>>
>>
>



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.


-----Original Message-----
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-
>>
>>
>>
>>
>