# Semiprime analogue of A059316

Maximilian Hasler maximilian.hasler at gmail.com
Wed Nov 28 19:50:33 CET 2007

```you should precise: ..."at least" n semiprimes
since for ..."exact" n semiprimes
your a(5) is not correct (you did not mark 22)
and as it stands the wording might be considered as ambiguous.
Also, for the sake of clarity, you should add "including endpoints".
All of these of course also apply to the existing sequence.

Maximilian

On Nov 28, 2007 2:16 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
> Semiprime analogue: Least integer m such that between m and 2m there
> are n semiprimes.
>
> 2,3,5,8,13,13,18,19,20,...
>
> Coincidence that first 5 values are successive Fibonacci numbers.
>
> n  a(n)  because
> 1  2     2,(4)
> 2  3     3,(4),5,(6)
> 3  5     5,(6),7,8,(9),(10)
> 4  8     8,(9),(10),11,12,13,(14),(15),16
> 5  13   13,(14),(15),16,17,18,19,20,(21),22,23,24,(25),(26)
> 6  13   same as above
> 7  18  21,22,25,26,33,34,35
> 8  19  21,22,25,26,33,34,35,38
> 9  20  21,22,25,26,33,34,35,38,39
>

> From seqfan-owner at ext.jussieu.fr  Wed Nov 28 19:16:21 2007
> Date: Wed, 28 Nov 2007 10:16:09 -0800
> From: "Jonathan Post" <jvospost3 at gmail.com>
> To: "Sequence Fans" <seqfan at ext.jussieu.fr>,
>         "jonathan post" <jvospost2 at yahoo.com>
> Subject: Semiprime analogue of A059316
>
> Is this right so far?
>
> A059316  Least integer m such that between m and 2m there are n primes.
>
> "between" here seems to include endpoints.
>
> Semiprime analogue: Least integer m such that between m and 2m there
> are n semiprimes.
>
> 2,3,5,8,13,13,18,19,20,...
>
> Coincidence that first 5 values are successive Fibonacci numbers.
>
> n  a(n)  because
> 1  2     2,(4)
> 2  3     3,(4),5,(6)
> 3  5     5,(6),7,8,(9),(10)
> 4  8     8,(9),(10),11,12,13,(14),(15),16
> 5  13   13,(14),(15),16,17,18,19,20,(21),22,23,24,(25),(26)
> 6  13   same as above
> 7  18  21,22,25,26,33,34,35
> 8  19  21,22,25,26,33,34,35,38
> 9  20  21,22,25,26,33,34,35,38,39

Following Max's argument, there are probably 4 different sequences (-1 below
means I don't know, larger than 1000 or not existing), for n=1,2,3,...:

2, 3, 5, 8, 15, 13, 18, 19, 20, -1, 29, 31, 33, 43, 44, 46, 47, 48, -1, 61, 62,

2, 3, 5, 8, 13, 13, 18, 19, 20, 29, 29, 31, 33, 43, 44, 46, 47, 48, 61, 61, 62,

3, 5, 8, 8, 13, 18, 18, 20, 20, 29, 30, 32, 43, 44, 44, 47, 48, 48, 61, 62, 67,

3, 5, 9, 8, 13, -1, 18, 21, 20, 29, 30, 32, 43, -1, 44, 47, 49, 48, 61, 62, 67,

The a(n) for the "at least" case is necessarily smaller or equal to the
"exact" case.

In ugly hacked Maple :

end:

A := proc(n,exa,withend)
end:

print("exact n, with end points") ;

print("at least n, with end points") ;

print("at least n, without end points") ;

print("exact n, without end points") ;
# end Maple

A Maple implementation of calculating the ordinary generating function
for sequences with linear homogeneous recurrences with constant coefficients
(and some very simple inhomogeneous cases) is in
http://www.strw.leidenuniv.nl/~mathar/progs/GenFLinRec.mp .
A short introduction on this is in
http://www.strw.leidenuniv.nl/~mathar/public/mathar20071126.pdf .
No new mathematics is involved; it is merely useful if one wishes to compile
ordinary generating function on some industrial scale.
--
Richard J Mathar                Tel (+31) (0) 71 527 8459
Sterrewacht Universiteit Leiden Fax (+31) (0) 71 527 5819
Postbus 9513
2300 RA Leiden                  E-mail mathar at strw.leidenuniv.nl
The Netherlands                 URL http://www.strw.leidenuniv.nl/~mathar

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