[seqfan] Re: Help with a(n) and a cumulative sum

[David Seal]

With regard to:

> The question is, do we know there is an infinite one that begins
> 1,2,3,5,4,6,7 (say)?  If so, then we would know the first 7 terms for
> certain. ...

Would we? Or would we only know them on the assumption that the sequence exists at all?

E.g. consider the description "the lexicographically earliest strictly increasing sequence of strictly positive integers that is not the sequence defined by a(n) = n", we know that there is a strictly increasing sequence of strictly positive integers that is not the sequence defined by a(n) = n which starts 1,2,3,4,5,6,7, for example the sequence defined by a(n) = n if n <= 7, a(n) = n+1 otherwise, and we also know that the first seven terms of any such sequence cannot be lexicographically earlier than 1,2,3,4,5,6,7.

But that doesn't establish that the first seven terms of the sequence described by "the lexicographically earliest strictly increasing sequence of strictly positive integers that is not the sequence defined by a(n) = n" are 1,2,3,4,5,6,7, but only that if that description defined a sequence, it would start 1,2,3,4,5,6,7. And in fact that description does not define a sequence, because no matter which such sequence I give, I can find a lexicographically smaller such sequence by constructing one that deviates from a(n) = n at a later value of n.

David 


> On 15 July 2019 at 06:24 Neil Sloane <njasloane at gmail.com> wrote:
> 
> 
> There is a parallel discussion about this same sequence running on the
> math-fun mailing list.
> 
> I just sent this message to that list:
> 
> (quote)
> "Nothing deep here" I said, referring to the class of "lexicographically
> earliest sequences".
> I take it back. A282317, the lex. earliest cube-free binary sequence,
> needed an argument from
> topology to show that it exists.
> 
> So A309151 may not be so trivial.
> 
> The present definition of A309151 is  "Lexicographically earliest sequence
> of distinct terms starting with a(1) = 1 such that a(n) doesn't share any
> digit with the cumulative sum a(1) + a(2) + a(3) + ... + a(n-1) + a(n)"
> 
> But this is the finite sequence 1,2,3,4, no? It can't be extended! It
> satisfies the conditions, and any other sequence satisfying the conditions
> must start 1,2,3,m with m >= 5.
> 
> Maybe the authors should modify the definition to say "Lexicographically
> earliest infinite sequence of distinct terms starting with a(1) = 1 such
> that a(n) doesn't share any digit with the cumulative sum a(1) + a(2) +
> a(3) + ... + a(n-1) + a(n)."? But then we don't know how many of the
> present terms are correct!  I'm sending the sequence back to the editing
> stack.
> (end quote)
> 
> Thanks to Robert Israel, we know there are infinite sequences with the
> property.
> The question is, do we know there is an infinite one that begins
> 1,2,3,5,4,6,7 (say)?  If so, then we would know the first 7 terms for
> certain.  Eric, do you know?
> 
> 
> Best regards
> Neil
> 
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
> 
> 
> 
> On Sun, Jul 14, 2019 at 11:53 PM Neil Sloane <njasloane at gmail.com> wrote:
> 
> > Robert,  I was just looking at A309151, as it happens.  Could you add your
> > sequence 1, 6, 4, 66, 34, 666, 334, ... to the OEIS please?  It is a useful
> > data point.
> >
> > Is A309151 going to need an argument from topology to show that it exists
> > (thus proving I was wrong when I said it was not deep!) ?  Jean-Paul?
> >
> > Best regards
> > Neil
> >
> > Neil J. A. Sloane, President, OEIS Foundation.
> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> > Phone: 732 828 6098; home page: http://NeilSloane.com
> > Email: njasloane at gmail.com
> >
> >
> >
> > On Sun, Jul 14, 2019 at 11:29 PM <israel at math.ubc.ca> wrote:
> >
> >> There are infinite sequences such that a(n) doesn't share any digit with
> >> the cumulative sum. For example: 1, 6, 4, 66, 34, 666, 334, ... with
> >> cumulative sums 1, 7, 11, 77, 111, 777, 1111, ... So your sequence
> >> A309151
> >> can indeed be infinite.
> >>
> >> Cheers,
> >> Robert
> >>
> >> On Jul 14 2019, Éric Angelini wrote:
> >>
> >> > Hello SeqFans, Here is the draft of a strange sequence:
> >> > https://oeis.org/draft/A309151 It says: Lexicographically earliest
> >> > sequence of different terms starting with a(1) = 1 such that a(n)
> >> doesn't
> >> > share any digit with the cumulative sum a(1) + a(2) + a(3) + ... +
> >> a(n-1)
> >> > + a(n). And in the Comments section: As this sequence needs a lot of
> >> > backtracking, we don't guarantee the accuracy of the last 79 integers
> >> of
> >> > the 1079-term b-file. Indeed, the problem comes from the fact that some
> >> > cumulative sums quickly block the extension of the sequence. This is
> >> the
> >> > case with 10 (or any other sum ending in zero). But this is the case
> >> too
> >> > with 301 after
> >> > 1,2,3,5,4,6,7,8,9,10,11,12,14,22,20,23,24,30,13,15,16,18,28. So, we
> >> > bumped quite often in a "bad sum" (on the average, the sequence was
> >> > extended by 100 terms for every backtrack). To make a prior list of
> >> "bad
> >> > sums" is difficult (meaning impossible, I guess): 258002 is such a "bad
> >> > sum" if you have previously used {1,3,4,6,7,9,39,49,69,79} else 258002
> >> +
> >> > 79 = 258081 would be ok. So my questions are: could the sequence be
> >> > infinite? Could a list of "bad sum numbers" be easely defined and used?
> >> > Post-scriptum: I am working, together with Jean-Marc, on the variants:
> >> > "Lexicographically [...] such that a(n) shares exacly k digit with the
> >> > cumulative sum a(1) + a(2) + a(3) + ... + a(n-1) + a(n)", with k = 1,
> >> 2,
> >> > 3, ... Best, É.
> >> >
> >> >--
> >> >Seqfan Mailing list - http://list.seqfan.eu/
> >> >
> >> >
> >>
> >> --
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >
> 
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