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

[Allan Wechsler]

David Radcliffe, that argument only works if the constraint has the
property that if all prefixes of S satisfy the property, then S itself
satisfies the property. I think that is just another way to say that the
set of all sequences that satisfies the property is closed (topologically).

I think that may be all we need. Note that David Seal's "counterexample" is
_not_ closed. Under Seal's constraint, the sequence 1,2,3,4,.... is the
limit point of a sequence of sequences (sorry!) where each sequence has the
desired property, but the limit doesn't. So Seal's set is not closed.

But Eric Angelini's constraint is OK: the set of sequences satisfying it
_is_ closed, so I think Radcliffe's argument goes through for Angelini's
sequence.

On Mon, Jul 15, 2019 at 10:08 PM David Radcliffe <dradcliffe at gmail.com>
wrote:

> 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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