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

[Allan Wechsler]

I'm not sure why the number of numbers with 11 letters is different in US
and UK usage. Although US usage is pretty standardized, there seem to be
several competing UK systems, having to do with whether the long scale or
the short scale is used (historically, the UK has been all long-scale, but
many publishers and writers are changing to short-scale), and if the long
scale is used, whether "milliard" is preferred to "thousand million", I
think all the UK customs want the word "and" after "hundred" if there is
any content below 10^2, while the US eschews the "and".

Anyway, I think A(11) only differs if we are using "milliard".

In the US, I think there are 31 numbers with 11 letters: [2389][378],
7[459], [459]00, [126]000, 10000, [459]000000, [459]000000000,
[126]000000000000. I probably could have missed some.

Do we use the Conway/Wechsler extended convention? (I vote no. That means
that number names stop at one unvigintillion minus one.)

On Sat, Apr 22, 2023 at 11:41 PM Olivier Gerard <olivier.gerard at gmail.com>
wrote:

> 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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> >
>
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