A371478 Numbers k for which k swaps are needed to bubble-sort the US English number name of k.

2, 17, 21, 25

Offset: 1

Author

Eric Angelini and Nicolas Graner, Mar 25 2024

Keywords

• base
• nonn
• word
• fini
• full

Comments

No swaps involve hyphens or spaces.

Only four such numbers < 100 have this property: are there more?

In French, only three such numbers are known: 1, 14, 23.

For number names up to one million, the number of swaps does not exceed 712. - Hans Havermann, Mar 25 2024

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

Examples

			2 is a term since "two" requires 2 swaps of adjacent letters to sort into ascending alphabetical order: TWO -> TOW -> OTW.
		

Links

• Eric Angelini and Nicolas Graner, More bubble sorting, personal blog, March 2024.
• Wikipedia, Names of Large Numbers

Crossrefs

Cf. A064038, A070196, A033860.