[seqfan] How many numbers have n letters?

[Neil Sloane]

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