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

[Olivier Gerard]

Neil,

I have this here on my bookshelf . There is a second volume. Same editor.

Although most puzzles in the book are not frankly about mathematics
and loaded/coded with british culture references, history, literature,
crossword wit and the like. Some are interesting, puzzling and hard.
Some are very poor jokes.  I hope for GB that spies there do not spend too
much time on this kind of games.

Olivier


On Sun, Apr 23, 2023 at 6:25 AM Neil Sloane <njasloane at gmail.com> wrote:

> 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
> >
> >
>
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