[seqfan] Re: Another surprising omission from OEIS
Andrew Weimholt
andrew.weimholt at gmail.com
Wed Nov 11 12:44:01 CET 2009
Sorry for beating this subject to death, but I don't think I
adequately articulated why I imposed x>0 in...
x*b^0 + x*b^1 + x*b^2 + ... + x*b^k, where x>0, b>x, k>1
One could argue that the x>0 restraint that I have put into the
formula is artificial and that excluding
this restraint allows 0 to be admitted into the sequence. What I was
really trying to capture, was that
the highest position of a non-zero coefficient for the powers of b is
is one way to define the length,
and in the definition of the sequence, length is required to be
greater than 2. So whether 0 is included
really comes down to how we define length. Is length the number of
symbols in a string? Or is it 1 plus
the highest power of b with a non-zero coefficient? (the latter leaves
the length of zero undefined)
Or something else? And if it is the number of symbols in a string, do
we permit leading zeros?
This sequence aside, there are several instances where it would be
more convenient to consider 0
to have 0 digits rather than 1 digit.
I'm still somewhat neutral on whether to include 0, and would like to
hear some other thoughts on this question.
Andrew
On Wed, Nov 11, 2009 at 12:17 AM, Andrew Weimholt
<andrew.weimholt at gmail.com> wrote:
> To be more precise, my example
>
> x*b^0 + x*b^1 + x*b^2 + ... + x*b^k, where x>0, b>1, k>1
>
> should be rewritten as
>
> x*b^0 + x*b^1 + x*b^2 + ... + x*b^k, where x>0, b>x, k>1
>
> Andrew
>
> On Wed, Nov 11, 2009 at 12:12 AM, Andrew Weimholt
> <andrew.weimholt at gmail.com> wrote:
>> On Tue, Nov 10, 2009 at 9:16 PM, <franktaw at netscape.net> wrote:
>>
>>> Should this sequence include 0 (000 in any base)?
>>
>> I'm not sure. It seems logical, but at the same time, a lot of
>> sequences would be different if we allowed an arbitrary number of
>> leading zeros. For instance, should 1000 be a considered a palindromic
>> number because it could be written 0001000? Conversely, should 111 be
>> disqualified as a repdigit because it could be written 0000111? The
>> fact that 000 can be used to represent 0 is an artifact of our syntax
>> for representing numbers. If we try to define the concept of a number
>> being a "repdigit with length >2 in some base" with more mathematical
>> rigor, we come up with something like this...
>>
>> Numbers that can be put in the form:
>> x*b^0 + x*b^1 + x*b^2 + ... + x*b^k, where x>0, b>1, k>1
>>
>> ...which disqualifies 000.
>>
>> That said, I still haven't convinced myself that 0 ought to be
>> excluded, because it depends on how we define this sequence and on the
>> definition of "repdigit". Does the term "repdigit" need to be defined
>> by a mathematical expression as the one above, or do we merely define
>> it as a string of symbols that can be understood to represent a
>> number? In the latter case, 000 qualifies.
>>
>> Anyone else want to weigh in before I submit the sequence?
>>
>> Andrew
>>
>
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