%I #26 Mar 26 2024 13:19:13
%S 2,17,21,25
%N Numbers k for which k swaps are needed to bubble-sort the US English number name of k.
%C No swaps involve hyphens or spaces.
%C Only four such numbers < 100 have this property: are there more?
%C In French, only three such numbers are known: 1, 14, 23.
%C For number names up to one million, the number of swaps does not exceed 712. - _Hans Havermann_, Mar 25 2024
%C There are no more terms. Computationally, there are none < 10^9. If the name of n has k letters, a(n) <= k*(k-1)/2 from the properties of bubble sort. Also, if 1000^(e-1) <= n < 1000^e, then k <= 41*e, where the loose upper bound comes from "threehundredseventythree" and "quattuordecillion" considering all terms up to 10^66 using English names of large numbers (see Wikipedia link). Thus, a(n) <= 820*e^2 <= 1000^(e-1) for e >= 3. Similar bounds can be derived for extended naming schemes. - _Michael S. Branicky_, Mar 26 2024
%H Eric Angelini and Nicolas Graner, <a href="https://cinquantesignes.blogspot.com/2024/03/more-bubble-sorting.html">More bubble sorting</a>, personal blog, March 2024.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Names_of_large_numbers">Names of Large Numbers</a>
%e 2 is a term since "two" requires 2 swaps of adjacent letters to sort into ascending alphabetical order: TWO -> TOW -> OTW.
%Y Cf. A064038, A070196, A033860.
%K base,nonn,word,fini,full
%O 1,1
%A _Eric Angelini_ and Nicolas Graner, Mar 25 2024