Base Rep Puzzle

[Jack Brennen]

Graeme McRae wrote:
> By dumb trial and error for n=2,3,... I get 1,2,9,81,1024,16384,1953125. 
> The way to find element n of this sequence is to make a list of powers 
> of 2..n, and then find the smallest power, x, such that 
> floor(ln(x)/ln(k)) is different for each k in 2..n.  This sequence is 
> not in OEIS.
> --Graeme

To continue the sequence, and listing them as powers (errors are
possible, I wouldn't trust these as 100% verified):

1^1
2^1
3^2
3^4
4^5
4^7
5^9
6^12
7^13
8^16
9^18
9^21
10^24
11^27
12^29
...

The base and exponent of the numbers in the sequence seem to follow
predictable trends.  The base might be n-O(log n), and the
exponent maybe n*O(log n) ???

The sequence of bases -- does it ever skip an integer?  Does it
ever go backward?

The sequence of exponents -- it seems safe to conjecture that it's
monotonically increasing.


Once I'm at my work computer, I can write a script to check these
conjectures up to a higher search limit than I can do by hand with
a spreadsheet.