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

[David Radcliffe]

Doesn't the following argument work?

Let S be the set of all infinite sequences of distinct positive integer
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).
Robert Israel has shown that S is a nonempty set.

Let S_n consist of the restrictions of the functions in S to {1, 2, ...,
n}. Each S_n has a lexicographically least element s_n, and if n > m then
s_n extends s_m. Therefore, the union of the s_n is the required sequence.

On Mon, Jul 15, 2019 at 2:57 PM jean-paul allouche <
jean-paul.allouche at imj-prg.fr> wrote:

> Dear Neil, dear all
>
> Yes it looks like a topological argument should make it. (But
> it is not really clear to me [yet] how to do this.)
>
> best wishes
> jean-paul
>
> Le 15/07/2019 à 05:53, Neil Sloane a écrit :
> > 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.
> >>>
> >>> --
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> >>>
> >>>
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