[seqfan] Re: Concatenate the sums of the neighboring digits

Neil Sloane njasloane at gmail.com
Tue Nov 5 07:23:13 CET 2019


David,  Since I simply copied the text from Hans's email, I share the blame.

How about if you change the text to make it correct, and at the end say
something along the lines of

"Argument corrected and completed by ~~~~."  ?

And thank you.  (I am just swamped with emails and other OEIS matters, and
I don't have time to do
as much checking as I would like, so I'm forced to accept a lot of what
people send me. )


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 Mon, Nov 4, 2019 at 10:45 PM David Seal <david.j.seal at gwynmop.com> wrote:

> Reading more of the sequences related to this problem, I noticed the
> following in the comments section of A328974:
>
> "Proof that this grows without limit, from Hans Havermann, Nov
>    01, 2019: (Start)
>
>  One way to prove that the trajectory of a number under
>    repeated application of the map defined in A053392 increases
>    without limit is to show that there exists a term containing
>    a strictly internal substring of three adjacent 9's.
>
>  Such a substring in term n will grow to four adjacent 9's in
>    term n+2; to six adjacent 9's in term n+4; to ten adjacent
>    9's in term n+6; ... to 2^k+2 adjacent 9's (see A052548) in
>    term n+2k, regardless of what happens in the rest of the
>    number.
>
>  In the present sequence a(41) contains two internal substrings
>    of three adjacent 9's. QED (End)"
>
> This proof is flawed, in that the assertion "Such a substring in term n
> will grow to four adjacent 9's in term n+2" is not true if the substring is
> followed by 0 and not preceded by a fourth 9. The smallest such number and
> its first two successors are 19990 --> 1018189 --> 1199917.
>
> It is fixable by replacing the second and third paragraphs by:
>
> "One way to prove that the trajectory of a number under
>    repeated application of the map defined in A053392 increases
>    without limit is to show that there exists a term containing
>    a non-final substring of three adjacent 9's.
>
>  If such a substring in term n is followed by a 0, it will grow
>    to a non-final substring of three adjacent 9's followed by a
>    1 in term n+2: *9990* --> *18189* --> *99917*
>
>  If such a substring in term n is followed by a digit d that is
>    neither 0 nor 9, it will grow to a non-final substring of
>    four adjacent 9's in term n+2: *999d* --> *18181(d-1)* -->
>    *9999d*
>
>  If such a substring in term n is followed by another 9, then
>    it is actually a substring of k 9's for k >= 4, which will
>    grow to a substring of (2k-3) 9's in term n+2: *[9]^d* -->
>    *[18]^(d-1)* -> *[9]^(2d-3)*
>
>  Putting those together, such a substring in term n will grow
>    to four adjacent 9's by term n+4; to five adjacent 9's by
>    term n+6; to seven adjacent 9's by term n+8; ... to 2^k+3
>    adjacent 9's (see A062709) by term n+4+2k, regardless of
>    what happens in the rest of the number."
>
> I am uncertain about the etiquette in such situations - should I just add
> the fixed proof to the comments, leaving the flawed proof as it is, or
> should I edit the flawed proof to correct it?
>
> David
>
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> Seqfan Mailing list - http://list.seqfan.eu/
>



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