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

[John Mason]

Olivier wrote :"I hope for GB that spies there do not spend too much time
on this kind of games. "
Actually, they spent most of the time passing information to the URSS.
https://en.wikipedia.org/wiki/Cambridge_Five

On Sun, Apr 23, 2023 at 5:41 AM 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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