inconsummate numbers

[John Conway]

On Wed, 17 Jan 2001, David W. Wilson wrote:

> What's interesting is the complement of this sequence, the "panconsummate"
> numbers (consummate in all bases >=2).  These are all the panconsummates
> I found <= 358000:
>
> 1 2 3 4 5 6 7 8 9 10 11 12 14 15 18 20 21 23 24 31 34 36 37 39 40 43 45
> 53 54 57 59 61 69 72 73 77 78 81 85 89 91 121 127 144 166 169 211 219
> 231 239 257 267 271 331 337 353 361 413 481 523 571 661 721 1093 1291 3097
>
> and I highly suspect that these are all there are, period.

    Good!  I wondered whether this sequence would be finite, and am
glad to see that it probably is.  Now I'm wondering about how to
prove it.

   Idea:  there's a fairly rich bunch of inconsummate numbers just
above  B^2/2; we might be able to find enough to make them cover
everything beyond some point, and search up to that point.  How
far did you go?

   John Conway