What number comes next?

series/number.name.lengths New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

What number comes next?

10, 4, 3, 11, 15, 14, ..., ?

There may be several possibilities.

-- Gordon W. Hauta <ghhhauta@sednet.mcd.on.ca>, February 1995

Show solution

Let's write them out in English:

   ten
   four
   three
   eleven
   fifteen
   fourteen

So the next one can at least be

   seventeen

If hyphens don't contribute to the count, then the following are also
legal after 14:

   21, 22, 26, 31, 32, 36, 44, 45, 49, 54, 55, 59, 64, 65, 69,
   81, 82, 86, 91, 92, 96

And no larger numbers are legal ("one hundred" is the English for
100).  If hyphens do contribute, then the following would be legal
after 14:

   41, 42, 46, 51, 52, 56, 61, 62, 66.

ObPuzzle:  Let's make the sequence unique by picking the smallest
number with the relevant length in English (and let's exclude hyphens
and spaces from the count).  I claim the sequence starts:

   1, 4, 3, 11, 15, 13, 17, 24, 23, 73, 3000, 11000, 15000, 101,
   104, 103, 111, 115, 113, 117, 124, 123, 173, 323, 373,

(i) Am I right?  (ii) What is the next number (it's less than 3073)?
(iii) I know of these counting words; how are the gaps filled?

   thousand, million, billion, trillion, quadrillion, quintillion,
   sextillion, septillion, octillion, nonillion, decillion,
   undecillion, dodecillion, ..., vigintillion, ..., centillion

(iv) Are there numbers for all lengths up to the length of the largest
number we have a systematic English name for?  Note: you can attempt
this puzzle using the English convention (in which <n>illion =
10^(6n)) or the American convention (in which <n>illion = 10^(3n+3)).

(v) Is the number with the longest word in either system given by
373,373,373,...,373?

-- Gareth Rees <gdr11@cl.cam.ac.uk>, February 1995

[ Editor: no answers to the ObPuzzle were found in the collected
postings. ]