Aronson's sequence

I was made aware of Aronson's sequence by Greg Ross' Futility Closet article on it three weeks ago. A couple of things caught my eye. The first was his use of "nine billion one million second" to example the "few T-less ordinals" that "don’t arrange themselves to mop up all the incoming Ts". It would have been a little more compelling if 9001000002 was actually in the sequence — which it is not. The closest t-free ordinal that is is 9001000702.

The second thing was Greg's "We had supposed that the sentence would end with … letter in this sentence. But an infinite sentence has no end..." English number names have been well-defined only up to 10^66-1 — although I fully expect (once Mathematica debugs its IntegerName function) that that will go up to 10^306-1. There exists a longer realization but it may take some time for Mathematica to decide to incorporate it and it isn't obvious to me if the machine-generated naming scheme is potentially infinite. All strictly increasing, current English number-name sequences are necessarily finite, whether or not it is so recognized.

Words from numbers

Last week I presented a "word" continuation puzzle. The algorithm used to create the list isn't too difficult to discover, applying English number words (one, two, three, ...) to the previous term (the zeroth assumed to be an empty string). Thus, one letter at a time, the second term from the first:

one +two
onet
onetw
netw

A letter gets added to the right if it doesn't already exist in the evolving string. It is deleted from the string if it does already exist. Thus the string will never contain more than one copy of any particular letter. If you noticed the double-comma near the end of my puzzle, that wasn't a typo: ourihten +onehundrednineteen results in an empty string, which +onehundredtwenty yields ohurweny, which +onehundredtwentyone yields one. Using Mathematica's built-in dictionary and ignoring already encountered words (such as one at index 121), here is a list of English found in a deep continuation:

         1 one
     21240 visaed
     45660 fads
     57242 ado
    155868 woad
    171524 aide
    271966 ad
    337664 waned
    347660 audit
    413700 and
    423066 roads
    507504 wained
    537056 goads
    557924 aid
    615808 wad
    619808 wade
    635830 wand
   1152766 mad
   1250766 moaned
   1272524 maid
   1298168 made
   2710904 maned
   3526644 mashed
  10984236 mawed
  16170624 maiden
  21730304 mated
  67092006 mead
 509056060 remands
 540798800 moated
1000080796 boards
1000146526 bards
1000152766 bad
1000298168 bade
1000530740 baud
1000558076 broads
1000562062 brands
1000748080 bandit
1000750040 band
1000816952 bandy
2000710904 baned
...

Why would all of our subsequent English dictionary words appear at even indices?

What's next?

one, netw, nwhre, nwhefou, nwhouiv, nwhouvsx, whoux, wouxeigt, wouxgt, wouxgen, wouxglv, ouxgt, ouxghirtn, xghif, xghftn, ghfsi, ghfivt, fven, fvit, fviwenty, fvitone, fviy, fviwnthre, vihtyou, houwntf, houfetysix, houfixwtv, oufxvnyg, oufxvgwi, oufxvgwhry, ufxvgwine, ufxvgnehryto, ufxvgnoihre, xvgneyr, xgnhf, gnfrys, gfhiv, fvryeiht, fvti, viory, viftone, vinerytwo, vinwfthre, vinwheyfur, nwhuotf, nwhurysix, whuixfotv, wuxvryegt, wuxvgfoi, wuxvgofty, wuxvgifne, uxvgnefyo, uxvgnoifhre, xvgnhety, xgnhf, gnhftys, ghifv, vfyei, vfti, vfsxy, vfitone, vfnesxytw, vfnwithre, vnwhesxyou, nwhoutf, nwhoufy, whoufixtv, woufvsyeigt, woufvgxi, woufgxisnty, wufgxivne, ufgxiseyo, ufgxiovnhre, gxihsety, gxhnf, ghfvtyi, ghfiv, fsnye, fvti, fveghy, fvitone, fvnghytw, fvnwithre, vnwgyou, nwouhtf, nwoufegysx, woufxihtv, woufxvyiht, woufxvgi, woufxvgety, wufxvgine, ufxvgnyo, ufxvgoinhre, xvghnty, xghnf, ghfnetys, ghfiv, fvyi, fvti, fvtiohur, fvtione, fvihunrewo, fviwnthre, viwtner, wtohunf, wtfnrsix, wtfixohuv, wfxvregh, wfxvgoui, wfxvgihrten, wfxgitoul, fxgihrv, fxgvouhrn, xgvourt, xgvhin, gvourst, gh, ourihten, , ohurweny, ?

Counting t-free ordinals

Yesterday I introduced written-out English ordinals that lacked the letter "t" and I asked how many there are less than "one vigintillionth". I have come to conclude that the total number of t-free ordinals may be expressed by (c+1)^x*o, where 'c' is the number of t-free cardinals less than 1000, 'x' is the number of t-free -illions through which we traverse, and 'o' is the number of t-free ordinals less than 1000.

c = 55   (1, 4, 5, 6, 7, 9, 11, 100, 101, 104, 105, 106, 107, 109, 111, 400, 401, 404, 405, 406, 407, 409, 411, 500, 501, 504, 505, 506, 507, 509, 511, 600, 601, 604, 605, 606, 607, 609, 611, 700, 701, 704, 705, 706, 707, 709, 711, 900, 901, 904, 905, 906, 907, 909, 911)

x = 10   (10^6, 10^9, 10^15, 10^30, 10^33, 10^36, 10^39, 10^48, 10^51, 10^60)

o =  7   (2, 102, 402, 502, 602, 702, 902)

So we have 56^10*7 = 2123138423672799232.

The latest version of Mathematica has a built-in IntegerName function that does both cardinals and ordinals:

count1[ncard_] := 
 Length[Select[Range[10^(ncard - 1), 10^ncard - 1], 
   StringFreeQ[IntegerName[#, "Words"], "t"] &]]; 
m = {count1[1], count1[2], count1[3]}

{6, 1, 48}

... The number of t-free cardinals of 1-digit, 2-digit, and 3-digit base-ten numbers.

count2[nord_] := 
 Length[Select[Range[10^(nord - 1), 10^nord - 1], 
   StringFreeQ[IntegerName[#, "Ordinal"], "t"] &]]; 
s = {count2[1], count2[2], count2[3]}

{1, 0, 6}

... The number of t-free ordinals of 1-digit, 2-digit, and 3-digit base-ten numbers.

Do[If[StringFreeQ[IntegerName[10^(3*i), "Words"], "t"], 
  s = Join[s, m*Total[s]], s = Join[s, {0, 0, 0}]], {i, 21}]; s

{1, 0, 6, 0, 0, 0, 42, 7, 336, 2352, 392, 18816, 0, 0, 0, 131712, 21952, 1053696, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7375872, 1229312, 59006976, 413048832, 68841472, 3304390656, 23130734592, 3855122432, 185045876736, 1295321137152, 215886856192, 10362569097216, 0, 0, 0, 0, 0, 0, 72537983680512, 12089663946752, 580303869444096, 4062127086108672, 677021181018112, 32497016688869376, 0, 0, 0, 0, 0, 0, 227479116822085632, 37913186137014272, 1819832934576685056, 0, 0, 0}

... The number of t-free ordinals of 1-digit to 66-digit base-ten numbers. Finally:

Total[s]

2123138423672799232

IntegerName[%, "Words"]

two quintillion, one hundred twenty-three quadrillion, one hundred thirty-eight trillion, four hundred twenty-three billion, six hundred seventy-two million, seven hundred ninety-nine thousand, two hundred thirty-two

Interestingly, Mathematica has attempted to bridge the gap between the dictionary large-number names up to 10^63 (one vigintillion) and the next dictionary entry at 10^303 (one centillion):

Table[{i, IntegerName[10^i, "Words"]}, {i, 63, 306, 3}] // TableForm

 63 one vigintillion
 66 one unvigintillion
 69 one duovigintillion
 72 one trevigintillion
 75 one quattuorvigintillion
 78 one quinvigintillion
 81 one sexvigintillion
 84 one septenvigintillion
 87 one octovigintillion
 90 one novemvigintillion
 93 one trigintillion
 96 one untrigintillion
 99 one duotrigintillion
102 one trestrigintillions
105 one quattuortrigintillions
108 one quintrigintillions
111 one sextrigintillions
114 one septrigintillions
117 one octotrigintillions
120 one novemtrigintillions
123 one quadragintillions
126 one unquadragintillions
129 one duoquadragintillions
132 one tresquadragintillions
135 one quattuorquadragintillions
138 one quinquadragintillions
141 one sexquadragintillions
144 one septenquadragintillions
147 one octoquadragintillions
150 one novemquadragintillions
153 one quinquagintillions
156 one unquinquagintillions
159 one duoquinquagintillions
162 one tresquinquagintillions
165 one quattuorquinquagintillions
168 one quinquinquagintillions
171 one sexquinquagintillions
174 one septenquinquagintillions
177 one octoquinquagintillions
180 one novemquinquagintillions
183 one sexagintillions
186 one unsexagintillions
189 one duosexagintillions
192 one tresexagintillions
195 one quattuorsexagintillions
198 one quinsexagintillions
201 one sesexagintillions
204 one septensexagintillions
207 one octosexagintillions
210 one novemsexagintillions
213 one septuagintillions
216 one unseptuagintillions
219 one duoseptuagintillions
222 one treseptuagintillions
225 one quattuorseptuagintillions
228 one quinseptuagintillions
231 one seseptuagintillions
234 one septenseptuagintillions
237 one octoseptuagintillions
240 one novemseptuagintillions
243 one octogintillions
246 one unoctogintillions
249 one duooctogintillions
252 one tresoctogintillions
255 one quattuoroctogintillions
258 one quintoctogintillions
261 one sexoctogintillions
264 one septenoctogintillions
267 one octoctogintillions
270 one novoctogintillions
273 one nonagintillions
276 one unonagintillions
279 one duononagintillions
282 one trenonagintillions
285 one quattuornonagintillions
288 one quinonagintillions
291 one senonagintillions
294 one septenonagintillions
297 one octononagintillions
300 one novenonagintillions
303 one centillions
306 one billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion

Notice the terminal "s" for 10^102 to 10^303. I've alerted Wolfram to the bug. Then, starting at 10^306, all hell breaks loose!

English t-free ordinals

second
one hundred second
four hundred second
five hundred second
six hundred second
seven hundred second
nine hundred second
one million second
one million one hundred second
one million four hundred second
one million five hundred second
one million six hundred second
one million seven hundred second
one million nine hundred second
four million second
four million one hundred second
four million four hundred second
four million five hundred second
four million six hundred second
four million seven hundred second
four million nine hundred second
five million second
five million one hundred second
five million four hundred second
five million five hundred second
five million six hundred second
five million seven hundred second
five million nine hundred second
six million second
six million one hundred second
six million four hundred second
six million five hundred second
six million six hundred second
six million seven hundred second
six million nine hundred second
seven million second
seven million one hundred second
seven million four hundred second
seven million five hundred second
seven million six hundred second
seven million seven hundred second
seven million nine hundred second
nine million second
nine million one hundred second
nine million four hundred second
nine million five hundred second
nine million six hundred second
nine million seven hundred second
nine million nine hundred second
eleven million second
eleven million one hundred second
eleven million four hundred second
eleven million five hundred second
eleven million six hundred second
eleven million seven hundred second
eleven million nine hundred second
one hundred million second
one hundred million one hundred second
one hundred million four hundred second
one hundred million five hundred second
one hundred million six hundred second
one hundred million seven hundred second
one hundred million nine hundred second
one hundred one million second
one hundred one million one hundred second
one hundred one million four hundred second
one hundred one million five hundred second
one hundred one million six hundred second
one hundred one million seven hundred second
one hundred one million nine hundred second
one hundred four million second
one hundred four million one hundred second
one hundred four million four hundred second
one hundred four million five hundred second
one hundred four million six hundred second
one hundred four million seven hundred second
one hundred four million nine hundred second
one hundred five million second
one hundred five million one hundred second
one hundred five million four hundred second
one hundred five million five hundred second
one hundred five million six hundred second
one hundred five million seven hundred second
one hundred five million nine hundred second
one hundred six million second
one hundred six million one hundred second
one hundred six million four hundred second
one hundred six million five hundred second
one hundred six million six hundred second
one hundred six million seven hundred second
one hundred six million nine hundred second
one hundred seven million second
one hundred seven million one hundred second
one hundred seven million four hundred second
one hundred seven million five hundred second
one hundred seven million six hundred second
one hundred seven million seven hundred second
one hundred seven million nine hundred second
one hundred nine million second
one hundred nine million one hundred second

...

25000 indexed terms

There are 7 t-free ordinals less than 10^6, 392 less than 10^9, 21952 less than 10^15. How many are less than 10^63 (one vigintillion)?