## Thursday, July 11, 2024

### Integers, in alphabetical order

I have been preoccupied, since the beginning of July, about integers in alphabetical order. Claudio Meller wrote this preamble to the Sequence Fanatics discussion list (taken from a British game show called The 1% Club): "If all  the numbers from 1 to 20 are written out in words and put into numerical order and then reorganised into alphabetical order, which is the only number that stays in the same position?" Claudio had noticed that, replacing 20 with other integers up to 1000, "all the numbers up to 399 are kept in their position at least once". I replied: "Not 8 or 18". Well, Claudio was wrong about 8 but I was wrong about 18. Claudio pointed out that 18 shows up for 1 to 815.

It hadn't shown up for 1 to 815 for me because I was using a letter-by-letter alphabetical order (spaces and punctuation are ignored) while Claudio was using a word-by-word alphabetical order (spaces are not ignored, a space preceding any letter of the alphabet).

In a Wikipedia article on alphabetical order, the treatment of multi-word strings reiterates again the two-fold situation: word-by-word or letter-by-letter. What's new to me is that the latter may sometimes be called "dictionary order".

Mathematica has an AlphabeticSort function that appears to function word-by-word. I am going to assume that word-by-word is the alphabetical-order prevailing standard and, if anyone prefers letter-by-letter, they ought to spell it out.

We are looking for the start of the integers in alphabetical order (American English, by which we mean Mathematica's IntegerName[#,"Words"]). I will point out that IntegerName uses a non-standard hyphen (character code 8208) instead of the normal, generic hyphen (character code 45). I'll never understand why they did that, but I digress.

In American English all integer names start with one of e, f, n, o, s, or t, so the first term is necessarily "eight". Since we are working in alphabetical order (i.e., word-by-word), the next term should incorporate an additional word. Mathematica has integer names up to (but not including) 10^306 and the alphabetically first word that fits is "billion", so "eight billion" is our second term. As before, we are looking for an additional word that will turn "eight billion" into a new integer. Our third term will be "eight billion eight". Our fourth term will be "eight billion eight hundred".

I decided to Google my four terms and got one hit (the reason I didn't link to the hit is because the webpage doesn't have the Google snippet information on it):

For some reason somebody thought "eight billion eight million" was the fifth term, followed by "eight billion eight thousand", "eight billion eighteen". In my mind, extending the sequence with additional words is the way to proceed. If an additional word is not logically possible, then we consider replacing the final existing word:

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eight billion eight hundred eight million eight hundred eight thousand two hundred seventy-three
eight billion eight hundred eight million eight hundred eight thousand two hundred seventy-two
eight billion eight hundred eight million eight hundred eight thousand two hundred six
eight billion eight hundred eight million eight hundred eight thousand two hundred sixteen
eight billion eight hundred eight million eight hundred eight thousand two hundred sixty
eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-eight
eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-five
eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-four
eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-nine
eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-one
eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-seven
eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-six
eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-three
eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-two
eight billion eight hundred eight million eight hundred eight thousand two hundred ten
eight billion eight hundred eight million eight hundred eight thousand two hundred thirteen
eight billion eight hundred eight million eight hundred eight thousand two hundred thirty
eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-eight
eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-five
eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-four
eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-nine
eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-one
eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-seven
eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-six
eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-three
eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-two
eight billion eight hundred eight million eight hundred eight thousand two hundred three
eight billion eight hundred eight million eight hundred eight thousand two hundred twelve
eight billion eight hundred eight million eight hundred eight thousand two hundred twenty
eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-eight
eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-five
eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-four
eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-nine
eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-one
eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-seven
eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-six
eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-three
eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-two
eight billion eight hundred eight million eight hundred eight thousand two hundred two
eight billion eight hundred eight million eight hundred eighteen
...

I have made the primes in the above bold.* A much larger version of the list (102 MB) is here (because we lose the bold, a separate, numerical file of the alphabetically first 43628 primes is here). An indexed decimal integer version of the list (20 MB) is here.

* A couple of two-consecutive primes are shown. The first three-consecutive primes instance:

eight billion eight hundred eight million eight hundred eighteen thousand five hundred sixty-nine
eight billion eight hundred eight million eight hundred eighteen thousand five hundred sixty-one
eight billion eight hundred eight million eight hundred eighteen thousand five hundred sixty-seven