Seq: Natural numbers that cannot be written as a sum of one or more consecutive composites

[T. D. Noe]

sequence appears to have two fewer terms: 5 and 11 .
Return-Path: <noe at sspectra.com>
X-Ids: 168
Mime-Version: 1.0
Message-Id: <p06240805c48628cc8909@[192.168.1.43]>
In-Reply-To: <Pine.GSO.4.64.0806232040100.18308 at hilbert.math.ubc.ca>
References: <5542af940806231807v47001f60s8ba655a87c42266f at mail.gmail.com>
 <Pine.GSO.4.64.0806232040100.18308 at hilbert.math.ubc.ca>
Date: Mon, 23 Jun 2008 21:33:03 -0700
To: Robert Israel <israel at math.ubc.ca>, Jonathan Post <jvospost3 at gmail.com>
From: "T. D. Noe" <noe at sspectra.com>
Subject: Re: Seq: Natural numbers that cannot be written as a sum of one
 or  more consecutive composites
Cc: SeqFan <seqfan at ext.jussieu.fr>
Content-Type: text/plain; charset="us-ascii"
X-Greylist: IP, sender and recipient auto-whitelisted, not delayed by milter-greylist-4.0 (shiva.jussieu.fr [134.157.0.168]); Tue, 24 Jun 2008 07:55:40 +0200 (CEST)
X-Virus-Scanned: ClamAV 0.93/7418/Tue Jun 10 15:08:07 2008 on shiva.jussieu.fr
X-Virus-Status: Clean
X-Miltered: at jchkmail.jussieu.fr with ID 48608C5B.000 by Joe's j-chkmail (http : // j-chkmail dot ensmp dot fr)!
X-j-chkmail-Enveloppe: 48608C5B.000/69.30.22.11/smtpauth.easystreet.com/smtpauth.easystreet.com/<noe at sspectra.com>
X-j-chkmail-Score: MSGID : 48608C5B.000 on jchkmail.jussieu.fr : j-chkmail score : X : R=. U=. O=. B=0.270 -> S=0.270
X-j-chkmail-Status: Unsure
Return-Path: <Eric.Angelini at kntv.be>
X-Ids: 166
X-IronPort-Anti-Spam-Filtered: true
X-IronPort-Anti-Spam-Result: ApsEAEvCYUjZiN4m/2dsb2JhbACBW7Bg
Content-class: urn:content-classes:message
Return-Receipt-To: "Eric Angelini" <Eric.Angelini at kntv.be>
MIME-Version: 1.0
Content-Type: text/plain;
	charset="us-ascii"
X-MimeOLE: Produced By Microsoft Exchange V6.5
Subject: Add to n its second smallest non-divider. Loop
Date: Wed, 25 Jun 2008 13:06:56 +0200
Message-ID: <F05775148D000C4B9321A5113D53CB7DCAA5BB at KNTV-SERVER01.kntv.local>
X-MS-Has-Attach: 
X-MS-TNEF-Correlator: 
Thread-Topic: Add to n its second smallest non-divider. Loop
Thread-Index: AcjWs5SixHfTnJ54S12ovlOGnvBP1A==
From: "Eric Angelini" <Eric.Angelini at kntv.be>
To: "math-fun" <math-fun at mailman.xmission.com>, <seqfan at ext.jussieu.fr>
X-Greylist: IP, sender and recipient auto-whitelisted, not delayed by milter-greylist-4.0 (shiva.jussieu.fr [134.157.0.166]); Wed, 25 Jun 2008 13:08:37 +0200 (CEST)
X-Virus-Scanned: ClamAV 0.93/7412/Mon Jun  9 16:34:57 2008 on shiva.jussieu.fr
X-Virus-Status: Clean
X-Miltered: at jchkmail.jussieu.fr with ID 48622734.00C by Joe's j-chkmail (http : // j-chkmail dot ensmp dot fr)!
X-j-chkmail-Enveloppe: 48622734.00C/195.238.6.175/mailrelay002.isp.belgacom.be/mailrelay002.isp.belgacom.be/<Eric.Angelini at kntv.be>
X-j-chkmail-Score: MSGID : 48622734.00C on jchkmail.jussieu.fr : j-chkmail score : . : R=. U=. O=. B=0.101 -> S=0.101
X-j-chkmail-Status: Ham
Content-Transfer-Encoding: 8bit
X-MIME-Autoconverted: from quoted-printable to 8bit by idf.ext.jussieu.fr id m5PB8c2M042176


Hello MathFun & SeqFans,

Rule:

< Add to n its second smallest non-divider. Loop. >

Let's start with n = 7, for instance

Is 1 a divider of 7? yes
   2                 no
   3                 no --> then new n = 7+3 = 10

Is 1 a divider of 10? yes
   2                  yes
   3                  no
   4                  no --> then new n =10+4= 14

Is 1 a divider of 14? yes
   2                  yes
   3                  no
   4                  no --> then new n =14+4= 18

Is 1 a divider of 18? yes
   2                  yes
   3                  yes
   4                  no
   5                  no --> then new n =18+5= 23

... etc.

Sequence starting with 7 is: 7,10,14,18,23,...

Sequence starting with 1 is: 1,4,9,13,16,21,...
                             [not in the OEIS]

Sequence starting with 2 is: 2,6,11,14,18,23,...
                             [merges with "7-seq"]

Sequence starting with 3 is: 3,7,10,14,...
                             [merges with "7-seq"]

Sequence starting with 5 is: 5,8,13,16,...
                             [merges with "1-seq"]

etc.

We might map those sequences like this:


   1--4--9---13--16--21--25--28--32--37--40 ...
             |                   |   |
      5--8---+               29--+   |
                                     |
   2--6--11--14--18--23--26--30------+
             |           |           |
   3--7--10--+           |       33--+
                         |
             12--19--22--+
                 |       |
             15--+       |
                         |
                 17--20--+

                         24--31--34--38--42 ...
                             |       |
                         27--+   35--+


What number starts the longest sequence containing 2008?
Best,
E.

---

P.-S.

This rule is not very productive:
< Add to n its first smallest non-divider. Loop. >

What about:
< Add to n its third smallest non-divider. Loop. >
Etc.