Fwd: No string of digits divisible by k

[Robert Israel]

... and similarly to have no substring of digits divisible by k,
where gcd(k,10) = 1, the integer can't have k digits.  For if
you consider the numbers A(m) consisting of the last m digits
for 1 <= m <= k, either all A(m) mod k are distinct, in which
case one must be == 0 mod k, or two (say A(m) and A(n) with
m < n) are in the same residue class mod k, in which case the
number consisting of the digits in A(n) but not A(m) is divisible
by k.

Cheers,
Robert

On Tue, 12 Jun 2007, Maximilian Hasler wrote:

> ---------- Forwarded message ----------
> From: Maximilian Hasler <maximilian.hasler at gmail.com>
> Date: Jun 12, 2007 2:20 PM
> Subject: Re: No string of digits divisible by k
> To: Eric Angelini <Eric.Angelini at kntv.be>
>
>
> Eric,
> in case you intend to submit this, consider rephrasing the 
> definition
> in a way making it a clear and precise definition:
> "having a string of digits divisible by 3" is very ambiguos
> 1/ the only digits divisible by 3 are 0,3,6,9
>
> 2/ zero is divisible by 3 so zeroless is useless
> (btw I'm not sure if this word exists)
>
> 3/ a "string" divisible by 3 ? Strings are not numbers ! A string 
> may
> be divided (split up) in substrings...
>
> 4/ I think you mean "substring"
>
> 5/ The sequence IS finite and there are no terms with more than 2
> digits: the remainder of a(n) divided by 3 is 1 or 2, and the same 
> is
> true for the first and last digit ; so removing either one or both 
> of
> these digits, you get a "substring" divisible by 3.
>
> Maximilian
>
> On 6/12/07, Eric Angelini <Eric.Angelini at kntv.be> wrote:
>> 
>> Hello SeqFans,
>> could someone examine this sequence:
>> 
>> - Zeroless integers having no string of digits divisible by 3.
>> 
>> I think it starts like this:
>> 
>> 1,2,4,5,7,8,11,14,17,22,25,28,41,44,47,52,... (not OEIS)
>> 
>> ... but I'm quite sure the sequence is finite. Example:
>> 
>> 715 will not fit because "15" is a substring divisible by 3;
>> 716 neither because "6" is divisible by 3,
>> 717 neither because "717" is divisible by 3;
>> 718 neither because of "18";
>> 719 neither because of "9";
>> 720 neither (has a "0")
>> etc.
>> 
>> If we replace the definition by:
>> 
>> - Zeroless integers having no string of digits divisible by k.
>> 
>> ... what could be the smallest k leading to an infinite sequence
>> -- if such k exists -- or to the longest finite sequence?
>> 
>> Sorry if this is hold hat.
>> 
>> Best,
>> É.
>> 
>> 
>> 
>> 
>> 
>> 
>