This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A371478 #26 Mar 26 2024 13:19:13 %S A371478 2,17,21,25 %N A371478 Numbers k for which k swaps are needed to bubble-sort the US English number name of k. %C A371478 No swaps involve hyphens or spaces. %C A371478 Only four such numbers < 100 have this property: are there more? %C A371478 In French, only three such numbers are known: 1, 14, 23. %C A371478 For number names up to one million, the number of swaps does not exceed 712. - _Hans Havermann_, Mar 25 2024 %C A371478 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 A371478 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 A371478 Wikipedia, <a href="https://en.wikipedia.org/wiki/Names_of_large_numbers">Names of Large Numbers</a> %e A371478 2 is a term since "two" requires 2 swaps of adjacent letters to sort into ascending alphabetical order: TWO -> TOW -> OTW. %Y A371478 Cf. A064038, A070196, A033860. %K A371478 base,nonn,word,fini,full %O A371478 1,1 %A A371478 _Eric Angelini_ and Nicolas Graner, Mar 25 2024