Identical-digit blocks in decimal representation of partition numbers

[Dean Hickerson]

Neil Sloane asked:

> Me:  Am I the only seqfan who finds this kind of investigation
> extremely repugnant?
>
> Good mathematics can be beautiful: this is not.

No, you're not.  I find almost all of the base-related sequences to be
completely worthless.  This is especially true of those that rely
specifically on base 10; why should certain sequences be in the database
just because humans have 10 fingers?

(Of course, if such a sequence has appeared in a publication then it
should be included.)

There are some base-related sequences that I consider at least moderately
interesting.  There are two types of these, those that refer to a specific
base in which the sequence is somehow different from the corresponding
sequences in other bases, and those that refer to all bases.

For example, there are several sequences in the OEIS related to palindromic
squares in various bases.  It's easy to see that there are infinitely many
such squares in every base >= 3, e.g. anything of the form 100...001^2 =
100...00200...001.  But in base 2 it's not so obvious, and so I find the
sequence (A003166) of numbers whose squares are palindromic in base 2
interesting.  (Does anyone know if there are infinitely many?)

For an example of the second type, we could define a(n) as the smallest
number whose square is palindromic and has at least 2 digits in base n.  That
turns out to be the same as the smallest positive integer whose square is
divisible by n+1, which is A019554 with a different offset.  We could also
ask for the smallest non-palindrome in base n whose square is palindromic,
or the smallest number whose square is a palindrome with an even number of
digits in base n.  (Do such numbers exist in all bases?)

I'm not planning to compute and submit these sequences, because I don't
think they're particularly important mathematically.  I.e. understanding
them isn't likely to lead to solutions of other problems, and they're not
likely to turn up in searches for 'naturally' occurring sequences.  But
at least they don't depend on an accident of human evolution.

Dean Hickerson
dean at math.ucdavis.edu