Balanced/Unbalanced numbers (and iterations)
This 2018 mail to SeqFan suddenly popped
up in my mailbox a few minutes ago, with a link to click — a link obviously not
safe.
Anyway, the question hereunder remains interesting, I guess (I do not know if there has been any
follow-up to this post on SeqFan as I was banned from the list a few years
ago. If yes, please accept my apologies, but I couldn't find anything in the OEIS).
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> Hello SeqFans,
> let's call
"Balanced" the integers of A036301 and "Unbalanced" the
others.
> [A036301 = "Numbers
n such that sum of even digits of n equals sum of odd digits of n."]
>
>Take an Unbalanced and add
to it its closest Balanced; if the result is Balanced, stop.
> If the result is
Unbalanced, iterate.
>
> Question:
> Do all Unbalanced end on
a Balanced?
>
> Example with the Unbalanced "1":
1 + 112 = 113
113 + 112 = 225
225 + 211 = 436
436 + 431 = 867
867 + 871 = 1738
1738 + 1744 = 3482
3482 + 3476 = 6958 is Balanced.
> Example with the Unbalanced "2":
2 + 112 = 114
114 + 112 = 226
226 + 211 = 437
437 + 431 = 868
868 + 871 = 1739
1739 + 1744 = 3483
3483 + 3489 = 6972
6972 + 6963 = 13935
13935 + 14003 = 27938
27938 + 27968 = 55906
55906 + 55820 = 111726
111726 + 111728 = 223454
223454 + 223449 = 446903
446903 + 446905 = 893808
893808 + 893813 = 1787621 is Balanced. (if I made no mistakes)
> P.-S. If there are two possible Balanced to be added to an Unbalanced (as this Unbalanced would stand at the same distance of the two Balanced) compute both branches.
>
> Best,
> É.
> Brussels time and day 19:29 Dec 7, 2018 — Seqfan Mailing list
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Next day update
Maximilian H. was quick to copy/paste to me the discussion the SeqFan list had 5 years ago. It started with the hereunder remark by Neil Sloane:
> On Sun, 9 Dec 2018, 11:40 Neil Sloane <...@gmail.com > wrote:
>
> I liked it a lot until I came to the point about "if there are two choices,
> follow both branches".
> So this is a non-deterministic process: you have to keep track of all the
> descendants - all the children, grandchildren, ... - until one of them
> finds a happy marriage and produces a balanced child?
Maximilian proposed then this new definition:
> Let's call "Balanced" the integers of A036301 <http://oeis.org/A036301>
> (Numbers n such that sum of even digits of n equals sum of odd digits of n.)
> and "Unbalanced" the others.
> Take an Unbalanced and add to it its closest Balanced;
> if the result is Balanced, stop.
> if the result is Unbalanced, iterate.
> Question:
> Do all Unbalanced end on a Balanced?
Then came Zak Seidov:
> (...) replace "closest" by "next larger"
Maximilian:
> ... good point!
... The challenge is now (June 28th, 2023) to answer this new question:
* Let's call "Balanced" the integers of A036301 <http://oeis.org/A036301>
* (Numbers n such that sum of even digits of n equals sum of odd digits of n.)
* and "Unbalanced" the others.
* Take an Unbalanced and add to the next larger Balanced;
* – if the result is Balanced, stop.
* – if the result is Unbalanced, iterate.
* Question:
* Do all Unbalanced end on a Balanced?
... any taker?-))
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(An unbalanced sketch by Tereza Lochmann)
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