[SeqFan]: Base-sequences. Was: A121760/1: two (interesting?) sequences

[Antti Karttunen]

Ralf Stephan wrote:

>I agree those are not base but wasn't there a rule to only
>include the first five of sequences containing an arbitrary
>number (OK the first ten or more for really important ones)? 
>
>  
>
>>    A039303, Number of distinct quadratic residues mod 6^n. 	
>>    A039304, Number of distinct quadratic residues mod 7^n. 	
>>    A039305, Number of distinct quadratic residues mod 8^n. 
>>    A039306, Number of distinct quadratic residues mod 9^n. 	
>>    A000993, Number of distinct quadratic residues mod 10^n = number of
>>             distinct n-digit endings of base 10 squares. 
>>
>>Maybe also:
>>
>>    A008975	Pascal's triangle mod 10
>>    A074822	Primes p(n) such that p(n) + 4 = p(n+1) and p(n) == 9 (Mod 10)
>>    A087355	n^10 mod 10^n
>>    
>>

Well, if one collects all the sequences with an integer parameter k into 
a nicely
organised table (with links to both directions) and/or index-entry, then 
I think we
should allow larger valus of k as well.
E.g. I remember that some time ago Creighton Dement encountered
the sequence http://www.research.att.com/~njas/sequences/A068038
in connection with his Floretion-related sequences.
However, it does not occur until as the _seventeenth_ row of the table 
A068009.

But back to bases:
Franklin T. Adams-Watters (franktaw at netscape.net) wrote August 22 2006:

>"When base 10 is implicit and we fumble with the digits to invent 
>something" is practically a description of the worst kind of base sequence.
>  
>My proposed redefinition of the base keyword is "a sequence which, at 
>some point in the definition, treats numbers as numerals (strings)".
>  
>  
>
The problem here is that there are not any hard mathematical definition 
for "when numbers
are treated as strings and when not", as all such functions can be 
represented as (primitive, mostly)
recursive functions, as any other mathematical functions (most of them, 
in any case) operating on
integers.

>There are a couple of borderline cases: if we are looking only at the 
>final digit (or the digital root), is that base, or just modulus?
>
Also, when taking a "digital root" in base n,
(see e.g. http://www.research.att.com/~njas/sequences/A010888 )
or when taking "alternative digital root" (i.e. alternatively add and 
subtract the digits)
then we are actually just doing something simple related to modulo n+1
or modulo n-1. (c.f. the well-known divisibility algorithms for 9 and 11).


Antti