[seqfan] Re: Very nice new sequence A329126 [1, 6, 42, 60, 139810, 126, ...]

[Allan Wechsler]

There's a clear pattern here, and I wonder if it persists. Express n in the
form 2^k * d, where d is odd. Then (conjecture) the solution is 1 (0^(d-1)
1)^(d-1) 0^k. This fits all the examples given, and seems plausible.


On Tue, Nov 12, 2019 at 10:32 PM Neil Sloane <njasloane at gmail.com> wrote:

> A new sequence sent in by Alon Ran, A329126,
> which starts 1, 110, 101010, 111100, 100010001000100010, 1111110,
>
> a(n) is defined to be the lex. earliest string of numbers such that when
> read in every base b >= 2, a(n) is divisible by n.
>
> It follows from the definition that all the digits are 0 and 1.  The author
> gives a number of other remarks, but it was not completely clear to me
> which were empirical observations and which were theorems. I do not have
> time to work on it, but it looks like a lovely problem. He gives 21 terms.
>
> The strings are huge. If they are rewritten in base 10 you get A329000
> = 1,6,42,60,139810,126,..., which makes it a lot easier to remember the
> first few terms.
>
> I hope someone will look into this and figure out what is going on.
>
> It has keyword "base", but it does not depend on any one base - it depends
> on all bases (so maybe that keyword is not justified).
>
> Neil
>
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