Suppose we count upward from 1 in the usual manner:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 . . .

Now let us substitute for each number above its standard name written out in English. (I have capitalized and bolded the initial letters, as they momentarily will become the focus of our attention.)

One Two Three Four Five Six Seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen . . .

Let us now make a list consisting only of those initial letters. Its first 100 elements are shown here:

O T T F F S S E N T E T T F F S S E N T T T T T T T T T T T T T T T T T T T T F F F F F F F F F F F F F F F F F F F F S S S S S S S S S S S S S S S S S S S S E E E E E E E E E E N N N N N N N N N N O . . .

The discussion henceforth will relate to this last sequence.

It is evident from a cursory glance that within the above string there are many instances where the same letter occurs several times in a row: near the beginning there are two consecutive Ts, then two consecutive Fs, then two consecutive Ss, and so forth. Go just a little further, and there's a run of 20 consecutive Ts (as a reminder, that would reflect the fact that the spelled-out English name of each and every integer from 20 through 39 begins with the letter T).

The astonishing coincidence I found involves the positions within the sequence where certain long strings of identical consecutive initial letters occur . . . .

Let us go through the sequence, starting from the beginning and always moving forward. As we go, let us do three things:
• make a note of any complete string of consecutive identical letters we encounter that is longer than any that has previously occurred (note: it must be longer; one that is just equally long doesn't count!)
• each time we find such a string, record the position within the sequence where it starts (for example, that T T right near the beginning starts with the 2nd letter of the overall sequence, so its starting position is 2)
• each time we find such a string, also record the length of that particular string of identical letters (for example, that T T right near the beginning has exactly 2 Ts, so its length is 2).

I submit that there is no logical reason that those last two items — the starting positions of record-length strings, and the lengths of those strings, should have any relationship whatsoever to each other. For instance, consider what would happen if instead of the foregoing we were to record all strings of consecutive identical letters in the list; that is, if we were to eliminate the requirement that only those strings that are longer than all previous ones would count. We would then get this uninteresting, more-or-less random-looking description:
a "string" of Os, length 1, starting at position 1
a string of Ts, length 2, starting at position 2
a string of Fs, length 2, starting at position 4
a string of Ss, length 2, starting at position 6
a "string" of Es, length 1, starting at position 8
a "string" of Ns, length 1, starting at position 9
a "string" of Ts, length 1, starting at position 10
a "string" of Es, length 1, starting at position 11 . . .
. . . and so on.

But now let's put that requirement back in: the one where we record only those strings of consecutive identical letters that are longer than any previous such strings. Is there some logical reason that taking that particular step should cause the positions of those strings and their lengths to become related in some way? Why should that make any difference? I for one cannot suggest any reason why it should. Nevertheless, let's do it, and see what we get. . . .

a "string" of Os, length 1, starting at position 1
a string of Ts, length 2, starting at position 2
a string of Ts, length 20, starting at position 20
a string of Os, length 100, starting at position 100
a string of Ts, length 200, starting at position 200
a string of Os, length 1000, starting at position 1000
a string of Ts, length 2000, starting at position 2000
a string of Ts, length 20,000, starting at position 20,000
a string of Os, length 100,000, starting at position 100,000
a string of Ts, length 200,000, starting at position 200,000
a string of Os, length 1,000,000, starting at position 1,000,000
a string of Ts, length 2,000,000, starting at position 2,000,000
a string of Ts, length 20,000,000, starting at position 20,000,000
a string of Os, length 100,000,000, starting at position 100,000,000
a string of Ts, length 200,000,000, starting at position 200,000,000
a string of Os, length 1,000,000,000, starting at position 1,000,000,000 . . .
. . . and so on. This pattern holds perfectly, from beginning to end, for the entire sequence of counting numbers with respect to their conventional names in English, all the way up to and including . . .
a string of Ts, length 200 vigintillion, starting at position 200 vigintillion!