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

[Frank Adams-watters]

That's what the "base" keyword is for.

Franklin T. Adams-Watters


-----Original Message-----
From: David Seal <david.j.seal at gwynmop.com>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Thu, Nov 11, 2021 10:30 am
Subject: [seqfan] Re: Very nice new sequence A329126 [1, 6, 42, 60, 139810, 126, ...]

I suspect the current wording arose because the sequence isn't really a sequence of integers, but a sequence of representations of integers. E.g. if the sequence is read in base 2, it is the sequence of integers (represented in decimal):

1, 6, 42, 60, 139810, 126, ...

but if read in base 3, it is the sequence of integers (represented in decimal):

1, 12, 273, 360, 130754415, 1092, ...

and if read in base 4, it is the sequence of integers (represented in decimal):

1, 20, 1092, 1360, 17247241220, 5460, ...

etc. So "a(n) is the least positive integer which yields a multiple of n when read in any base" isn't accurate because you don't read integers in bases, but instead read digit strings in bases. But I agree that the current wording, while recognising that fact, made a worse mistake with "lexicographically earliest"...

I think it needs something like "the least positive integer whose decimal representation can be read in any numeric base (so consists of only 0s and 1s) and always yields a multiple of n when read in any numeric base".

David


> On 09/11/2021 06:34 Don Reble via SeqFan <seqfan at list.seqfan.eu> wrote:
> ...
> > %S A329126 1,110,101010,111100,100010001000100010,1111110
> > %N a(n) is the lexicographically earliest string of digits which
> >    yields a multiple of n when read in any numeric base.
> 
>     Does the author really mean "lexicographically earliest"?
>     There are many candidates for a(2): 110, 1010, 10010, 100010, ...,
>     Each is divisible by 2 in any base. Lexicographically,
>     each is earlier than the previous, and there is no earliest.
> 
>     I think he means simply "the least positive integer which yields
>     a multiple...".
> ...

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