n for which A108954(n) = Pi(2n)-Pi(n) is prime

[Olivier Gerard]

On Thu, Jul 31, 2008 at 00:29, Jonathan Post <jvospost3 at gmail.com> wrote:
> I don't know if this is worth submitting.  Comments?
>
> n for which A108954(n) = Pi(2n)-Pi(n) = the number of primes between n
> and 2n, inclusive, is prime.
>
> 4, 6, 7, 8, 9, 11, 13, 14, 16, 21, 23, 27, 28, 30, 31, 32, 33, 51, 53,
> 55, 56, 59, 60, 64, 65, 67, 68, 73, 74, 87, 88, 90, 96, 97, 98, ...
>

My personal opinion is : no, unless one at least of the various
links to papers (for instance the one you quoted in your post) uses or
analyses this subsequence.


> I just submitted a comment and hotlink for A108954, the link being to
>

Jonathan, Neil has just asked everyone *not to post Comments* until
the new Addition page is ready.

Please, just store your ideas for a while and let's discuss on seqfan
those requiring interaction with someone else than Neil when the time
has come.

Regards,

Olivier




ja> From seqfan-owner at ext.jussieu.fr  Thu Jul 31 08:10:02 2008
ja> Date: Thu, 31 Jul 2008 16:08:46 +1000
ja> From: Joerg Arndt <arndt at jjj.de>
ja> Subject: A025151 and A026826
ja> 
ja> This one
ja> A025151 Number of partitions of n into distinct parts >= 6.
ja> seems OK.
ja> 
ja> This one
ja> A026826 Number of partitions of n into distinct parts, the least being 5.
ja> has identical terms, it seems to be duplicate with wrong definition.
ja> 
ja> By definition the latter should be
ja> 
ja> (offset) 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7,
ja> 8, 10, 11, 13, 15, 17, 20, 23, 26, 30, 35, 39, 45, 51, 58, 66, 75, 84,
ja> 96, 108, 122, 136, 154, 172, 194

ftaw> From seqfan-owner at ext.jussieu.fr  Thu Jul 31 08:35:21 2008
ftaw> To: arndt at jjj.de, seqfan at ext.jussieu.fr
ftaw> Subject: Re: A025151 and A026826
ftaw> Date: Thu, 31 Jul 2008 02:34:12 -0400
ftaw> From: franktaw at netscape.net
ftaw> 
ftaw> These both look OK to me.  By their definitions, these sequences should
ftaw> be the same, except for initial terms and offset: add a part of size 5
ftaw> to a
ftaw> partition of n into distinct parts >= 6, and you get uniquely a
ftaw> partition of
ftaw> n+5 into distinct parts, with the least being 5.
ftaw> 
ftaw> I'm not sure how you got the sequence below.
ftaw> 
ftaw> Franklin T. Adams-Watters

These "partitions of n into distinct parts >= k" and
"partitions of n into distinct parts, the least being k-1" come in pairs
of similar, almost shifted but not identical, sequences,

A025147 (k=2)
A025148 (k=3)
A025149, A026824 (k=4)
A025150, A026825 (k=5)
A025151, A026826 (k=6)
A025152, A026827 (k=7)
A025153, A026828 (k=8)
A025154, A026829 (k=9)
A025155, A026830 (k=10)
I think they are basically correct. The fine point in the definitions is that
"distinct parts >= k" sets a lower bound to all parts, whereas "the least
being ..." means that the lower limit must be attained by one of the parts,
as Franklin argued. 

The (mild) problem that I see is that A025147 and A025148 have an offset of 0
which would mean "there is 1 partition of 0 into distinct parts >= 3," and this
is wrong.  As long as one extends the definition for A025147..A025155 by clinging
to the g.f. including the term 1*x^0, this is kind-of alright.
One ought perhaps split A025147 and A025148 into two different sequences each in
the manner it has been done for the higher k-numbers.

The clarifying strategy is to add explicit x-references (please check..):

%I A026824
%F A026824 a(n)=A025149(n-3), n>3. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A026825
%F A026825 a(n)=A025150(n-4), n>4. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A026825 G.f.: x^4*product_{j=5..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A026826
%F A026826 a(n)=A025151(n-5), n>5. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A026826 G.f.: x^5*product_{j=6..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A026827
%F A026827 a(n)=A025152(n-6), n>6. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A026827 G.f.: x^6*product_{j=7..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A026828
%F A026828 a(n)=A025153(n-7), n>7. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A026828 G.f.: x^7*product_{j=8..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A026829
%F A026829 a(n)=A025154(n-8), n>8. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A026829 G.f.: x^8*product_{j=9..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A026830
%F A026830 a(n)=A025155(n-9), n>9. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A026830 G.f.: x^9*product_{j=10..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A025149
%F A025149 a(n)=A026824(n+3). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A025149 G.f.: -1+product_{j=4..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A025150
%F A025150 a(n)=A026825(n+4). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A025151
%F A025151 a(n)=A026826(n+5). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A025151 G.f.: -1+product_{j=6..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A025152
%F A025152 a(n)=A026827(n+6). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A025152 G.f.: -1+product_{j=7..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A025153
%F A025153 a(n)=A026828(n+7). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A025153 G.f.: -1+product_{j=8..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A025154
%F A025154 a(n)=A026829(n+8). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A025154 G.f.: -1+product_{j=9..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

%I A025155
%F A025155 a(n)=A026830(n+9). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008
%F A025155 G.f.: -1+product_{j=10..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008