zeroless squares

[Pfoertner, Hugo]

-----Original Message-----
> From: David Wilson [mailto:davidwwilson at comcast.net] 
> Sent: Sunday, February 27, 2005 03:10
> To: seqfan at ext.jussieu.fr
> Subject: Re: zeroless squares
>
>
> The same naive estimate can easily be generalize to kth powers, giving the
> estimate s(d) = (10^d)^(1/k) - (10^(d-1))^(1/k) for d-digit kth powers.
> p(d) remains the same.  The resulting estimates have ratio
(9/10)*10^(1/k).
> We should expect an infinite number of zeroless kth powers when this ratio
> is >= 1, which it is for k <= 21.  For k >= 22, the ratio is < 1 and we
> should expect a finite number of zeroless kth powers.

This suggests a new sequence:

Smallest power with base>1, exponent>1 whose decimal representation doesn't
contain the digit 0:

base exponent  base^exponent

 2  1               2
 2  2               4
 2  3               8
 2  4              16
 2  5              32
 2  6              64
 2  7             128
 2  8             256
 2  9             512
 5 10         9765625
 3 11          177147
 3 12          531441
 2 13            8192
 2 14           16384
 2 15           32768
 2 16           65536
 4 17     17179869184
 2 18          262144
 2 19          524288


Hugo