[seqfan] Re: Another surprising omission from OEIS

franktaw at netscape.net franktaw at netscape.net
Thu Nov 12 19:23:41 CET 2009

There is no consistency to whether the base keyword is used for such 
"multi-base" sequences in the OEIS.  My impression (based on a 
considerable amount of looking) is that the majority of such sequences 
do use the base keyword.  I personally do use it for most such 
sequences, on the theory that people who are uninterested in base 
sequences are usually uninterested in sequences of this sort.

My rule for what is a base sequence is that any sequence which regards 
numbers as just numerals at some point in its computation is a base 
sequence, with the caveat that sequences that have other, equally 
simple definitions that do not fit this definition are not base 
sequences.  This means that base factorial and similar things should 
have the keyword.

Thus, by my standards, A007088 (numbers in base 2) correctly does not  
have the base keyword, since the alternative definition "numbers that 
are sums of distinct powers of 10" is equally simple.  By contrast, 
A007089 (numbers in base 3) correctly does have the keyword, since the 
corresponding definition -- numbers that are the sums of powers of 10 
with no number present more than twice -- is more complicated (it isn't 
even a comment in the sequence). Likewise, A000120 (number of 1's in 
binary expansion of n) correctly does not have the keyword, since the 
computation a(2n) = a(n), a(2n+1) = a(n) + 1 is just as simple; the 
ternary version (A053735) should -- and does -- have the keyword.

I don't claim that this matches the definition in the help file; but it 
matches the current usage at least as well as anything (again, usage is 
very inconsistent), and is more useful for people doing searches than 
any other definition I know of.

In this particular case, it is questionable whether the alternative 
definition -- numbers of the form c*sum(0<=k<=m, b^k) where b>1, m>1, 
c<b, and c>=0 (or c>0, if zero is not included) -- is as simple as the 
primary definition.  On balance, I would say that it is not, and thus 
this should be marked as a base sequence.

Franklin T. Adams-Watters

-----Original Message-----
From: Andrew Weimholt <andrew.weimholt at gmail.com>


By the way, my interpretation of the "base" keyword (Neil, correct me
if I am wrong) is that it applies to sequences which use a single base
in their definition (so that if you change the base, the sequence will
contain a different set of terms). Sequences which cover all the bases
are not really "base" sequences (which is why I didn't add the "base"
keyword to these submissions).
The OEIS help file for keywords says "base: dependent on base used for
sequence", which seems to support my interpretation.


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