n-digit numbers with n factors, each with a different number of digits

[Jonathan Post]

Dear Joshua,

I suggest that you eliminate the "base" aspect of this by considering the
array of it over all positive integer bases.  That is, T(k,n) is the
smallest composite number such that each of its factors has a different
number of digits in base k.

Then give us that array by antidiagonals, and comment on its main diagonal
T(n,n) and any other rows and columns and diagonals with nice formulae.

-- Jonathan Vos Post

On 12/5/06, Joshua Zucker <joshua.zucker at gmail.com> wrote:
>
> Based on a problem from a recent Mandelbrot competition, which asked
> something like:
> what is the smallest composite number such that each of its factors
> has a different number of digits?
>
> I thought it might be fun to start working on the sequence of numbers
> whose factors each have a different number of digits.
>
> But then I thought it might be even more fun to find the smallest
> n-digit number that has exactly n factors, each with a different
> number of digits.
>
> 1, 11, 121, 1111, 14641, 112211, 1771561, 11117777, 123187801,
> 1464143923, 25937424601, ...
>
> I'm of the opinion that this is interesting enough to share here but
> not interesting or useful enough to submit to OEIS, but if a couple
> people disagree then I'll work out some more terms and send it in.
>
> --Joshua Zucker
>
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