[seqfan] Re: How many numbers have n letters?

[Neil Sloane]

PS  "The most basic question" of course has an obvious answer: for any k,
there are only finitely many numbers with k letters, as long as we assume
that in the standard numbering, a name can only specify a single number.
So delete that question!

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com



On Sat, Apr 22, 2023 at 11:17 PM Neil Sloane <njasloane at gmail.com> wrote:

> Dear Sequence Fans, I've been going through a wonderful book of puzzles I
> came across the other day,
>
> GCHQ, The GCHQ Puzzle Book, Penguin, 2016.
>
>
>
> I found a bunch of new sequences which I have added to the OEIS (see
> A362120 onwards, and
>
> A362435 onwards). There were also many sequences already in the OEIS, and
> for these I added a reference to the book.
>
>
>
> There is one sequence I need help with, on page 123. The terms a(1) though
> a(10) are:
>
> 0, 0, 4, 3, 6, 6, 3, 13, 22, 35,
>
> and (my) definition is that a(n) is the number of [nonnegative / positive]
>
>
> numbers whose standard name in [British / American] English has n letters,
>
>
> or -1 if there are infinitely many numbers with n letters.
>
>
> So there are really four sequences. The only difference between
> nonnegative and positive
>
> is at n=4, where we get 3 for positive numbers (four, five, nine) or 4 for
> nonnegative numbers (include zero).
>
> Up though n=10 there is no difference between British and American
> English, according to GCHQ.
>
> The 35 numbers with ten letters are, according to the GCHQ web site,
>
>
>
> https://www.stephenpeek.co.uk/gchq_competitions/kristmas_kwiz/kristmas_kwiz_challenge_answers.pdf
>
>
> 24, 25, 29, 34, 35, 39, 43, 47, 48, 53, 57, 58, 63, 67, 68, 71, 72, 76,
> 84, 85, 89, 94, 95, 99, 100, 200, 600, 1000000, 2000000, 6000000, 10000000,
> 1000000000, 2000000000, 6000000000, 10000000000.
>
>
> The number of letters in n in the US is given by A005589, which has a
> modest b-file, and in the UK it is A362123, which has no b-file yet.
>
>
> I think the OEIS should have these four sequences, at least as far out as
> they can be reasonably well-defined.
>
>
> (There may be versions already in the OEIS, of course - I did not search
> very carefully.)
>
>
>
> But I don't even know the answer to the most basic question: what is the
> smallest k such that there are infinitely many numbers with k letters (in
> the standard numbering)?
>
>
>
> Here is a table of the number of numbers in the 11100-term b-file for
> A005589 with 1 through 40 letters:
>
> [0, 0, 4, 4, 6, 6, 3, 13, 22, 27, 22, 9, 15, 38, 63, 90, 100, 117, 199,
> 319, 399, 358, 235, 154, 153, 258, 364, 435, 539, 793, 1250, 1615, 1597,
> 1155, 582, 189, 27, 0, 0, 0]
>
>
> These questions must be well-studied!
> Best regards
> Neil
>
> Neil J. A. Sloane, Chairman, OEIS Foundation.
> Also Visiting Scientist, Math. Dept., Rutgers University,
> Email: njasloane at gmail.com
>
>