Responses interspersed below. On Wed, Jan 14, 2009 at 4:42 PM, Steve Witham <sw@tiac.net> wrote:

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From: "Allan Wechsler" <acwacw@gmail.com>

More "constructively", let a(0)=3, and then define a(n+1) as the smallest number whose name written out in English has a(n) letters.

If so, shouldn't a(1) = 1?

Ah, I think you have reconstructed Eric's original reason for explicitly specifying that the sequence be monotonic. If a(1)=1, the whole idea breaks down because there is no number whose English name has one letter.

6 seems to be the smallest a(0) that works with that definition (and I like that definition).

This works too.

The prefixes on English 'illions seem to be Latin names for numbers. If so, can we back-fit Latin transliterations of the English words to extend things indefinitely?

We trod this ground most heavily in a discussion on this list in the run-up to the publication of Conway's & Guy's *The Book of Numbers*. Coherent, consistent Latin names for numbers become hard to find somewhere around 1e6; above that point different writers used different systems. The Latin number names become inappropriate for use as prefixes long before that, unless you like names like unus-et-vigintillion. For TBoN, we presented a plausible extension (with minor corrections) of the system popularized by Chuquet; our system works up to the 999th zillion. For bigger numbers, we created a slightly-less-plausible extension of the extension; only by the tiniest margin did we avoid naming some poor number the "millivanillion". The 31-digit number I was groping for earlier fits within the Chuquet system just fine. A number whose name has that many letters is a different matter.