[seqfan] Re: Another surprising omission from OEIS
Jack Brennen
jfb at brennen.net
Thu Nov 12 22:52:58 CET 2009
Regarding 'base', I think a good test is whether the sequence is easily
defined without resorting to digit representations in some base.
For example:
Fibonacci numbers ending in 1, 3, 7, or 9 -- could be easily defined
as Fibonacci numbers coprime to 10. No need for 'base' keyword.
Fibonacci numbers not containing a '1' anywhere in their decimal
expansion. Should be 'base' -- can't be easily defined except
using decimal expansion.
(Neither sequence is in the OEIS, and I'm not proposing their
inclusion at this point...)
Note that digit representations are an artifact of our single-radix
positional notation system and variants thereof. There are certainly
many other ways to represent numbers -- Roman numerals for instance,
or the French spoken system, where 97 is stated as four-twenty-ten-seven
(quatre-vingt-dix-sept). I suspect this is why mathematicians tend to
look askance at 'base' sequences -- because they are an artifact of our
system for writing numbers, rather than a fundamental property of the
numbers themselves.
PS: There are only 31 Fibonacci numbers (including '0') which don't
contain a '1' in their decimal expansion. (Conjecture, of course.)
No other digit is so difficult to avoid. I suspect this is at
least partially explained by Benford's law, and partially by the
fact that 2/3 of all Fibonacci numbers are odd. (So the chance
of a random Fibonacci number starting with '1' is about 30%, and
the chance that it ends with '1' is about 13%.)
Leroy Quet wrote:
> With all the insults thrown at 'base' sequences in this thread, I want to defend base-2 'base' sequences.
> Unlike base-10, base 2 is more "natural", since it is often times the smallest-order base that is interesting.
>
> And then there are the applications of some base-2 sequences to computer science.
>
> Still, I may have inadvertently used the 'base' keyword for some of my binary-based sequences which are more simply defined by non-base means, a no-no according to an earlier post to this thread.
>
> Thanks,
> Leroy Quet
>
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