[seqfan] Re: A098786 incorrect (basic philosophy)
Marc LeBrun
mlb at well.com
Tue Mar 31 17:42:41 CEST 2009
>=Richard Mathar
> The philosophical question is: if the base-dependent sequences are
> made explicit by putting them into the database for a bunch of different
> bases, does that make them less offending to common taste (less
base-dependent,
> that is) ?
Our motive is try to avoid gratuitous or arbitrary sequences, not
base dependency per se. The "base" keyword is a symptom, not a diagnosis.
Consider a family of sequences that can all be generated from a
single parameterized function f[a,b,c,...](n) by taking different
specific bindings of the parameters a=A, b=B, c=C... We can't put
them ALL into the OEIS, so which of these qualify as "interesting"?
Certainly when some [A,B,C,...] result in sequences with unusual
properties that distinguish them from the rest of the ocean of other
choices then they are interesting. An example would be, say, the
Somos-K sequences for K=4, 5, 6 and 7.
But absent any such intrinsic distinctions then the sequences are
just arbitrary samples of the family. Having bunches of them doesn't
make any of them more interesting individually. Submitting the
"B=10" case as the representative of the family is often sufficient.
On the other hand it's worth checking if potentially interesting
things happen "near" the boundaries, typically for small parameter
values. Hence the rule of thumb, when submitting "B=10", to at least
look at "B=2" and perhaps a few other small values.
Of course once you've taken the trouble to accurately compute "2 or 3
lines" of values for several related sequences you might as well run
them by superseeker and then submit them too, for future reference
(and potential superseeker hits!)
Admittedly, there's no hard criteria. So-called "dismal arithmetic"
was constructed by abstracting binary algorithms and then setting
B>2. For B=2 it degenerates to regular arithmetic, but "nearby"
small cases, say B=3, 4, maybe 5, exhibit unique personalities and
might be included too. However B=10 isn't particularly different
from B=...8, 9 or B=11, 12... it's just convenient for expository
purposes (and of course to justify the silly name<;-).
I think all these considerations also apply to A067581. The B=10
entry represents the general idea, the B=2 case is degenerate, but
some "nearby" small cases, say B=3, 4, maybe 5, wouldn't be unwelcome
(and they might produce interesting "hits" in future superseeker
searches, always an important consideration!)
Moreover, the "baseness" of A067581 is intrinsic to the construction
of the sequence, and since there's nothing gratuitous mixed in I find
it an acceptable representative of a reasonably interesting family.
However if the definition said say, "smallest prime", instead of
"smallest integer" I'd find it boring, since the encoding of primes
(or Fibonacci numbers or whatever) in arbitrary bases has no
intrinsic interest or elegance to excite us.
Hope this ramble was helpful...
Best regards,
--Marc
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