It’s like watching a seed grow. You take a number and count how many 0s it contains, then how many 1s, how many 2s, 3s, 4s and so on. Then you create a new number by writing the count of each digit followed by the digit itself. Then you repeat the process with the new number.

Here’s how it works if you start with the number 1:

1

The count of digits is one 1, so the new number is this:

→ 11

The count of digits for 11 is two 1s, so the next number is:

→ 21

The count of digits for 21 is one 1, one 2, so the next number is:

→ 1112

The count of digits for 1112 is three 1s, one 2, so the next number is:

→ 3112

The count of digits for 3112 is two 1s, one 2, one 3, so the next number is:

→ 211213

What happens after that? Here are the numbers as a sequence:

1 → 11 → 21 → 1112 → 3112 → 211213 → 312213 → 212223 → 114213 → 31121314 → 41122314 → 31221324 → __21322314__…

That’s all you need, because something interesting happens with 21322314. The digit count is two 1s, three 2s, two 3s, one 4, so the next number is:

→ 21322314

In other words, 21322314 is what might be called a self-descriptive number: it describes the count of its own digits. That’s why I think this procedure is like watching a seed grow. You start with the tiny seed of 1 and end in the giant oak of 21322314, whose factorization is 2 * 3^2 * 13 * 91121. But there are many more self-descriptive numbers in base ten and some of them are much bigger than 21322314. A047841 at the *Online Encyclopedia of Integer Sequences* lists all 109 of them (and calls them “autobiographical numbers”). Here are a few, starting with the simplest possible:

22 → two 2s → 22

10213223 → one 0, two 1s, three 2s, two 3s → 10213223

10311233 → one 0, three 1s, one 2, three 3s → 10311233

21322314 → two 1s, three 2s, two 3s, one 4 → 21322314

21322315 → two 1s, three 2s, two 3s, one 5 → 21322315

21322316 → two 1s, three 2s, two 3s, one 6 → 21322316*

1031223314 → one 0, three 1s, two 2s, three 3s, one 4 → 10

31223314

3122331415 → three 1s, two 2s, three 3s, one 4, one 5

→ 3122331415

3122331416 → three 1s, two 2s, three 3s, one 4, one 6

→ 3122331416*

*And for 21322317, 21322318, 21322319; 3122331417, 3122331418, 3122331419.

And here’s what happens when you seed a sequence with a number containing all possible digits in base ten:

1234567890 → 10111213141516171819 → 101111213141516171819 → 101211213141516171819 → __101112213141516171819__…

That final number is self-descriptive:

__101112213141516171819__ → one 0, eleven 1s, two 2s, one 3, one 4, one 5, one 6, one 7, one 8, one 9 → __101112213141516171819__

So some numbers are self-descriptive and some start a sequence that ends in a self-descriptive number. But that doesn’t exhaust the possibilities. Some numbers are part of a loop:

__103142132415__ → 104122232415 → __103142132415__…

__104122232415__ → 103142132415 → __104122232415__…

__1051421314152619__ → 1061221324251619 → 1051421314152619…

__5142131415261819__ → 6122132425161819 → __5142131415261819__…

__106142131416271819__ → 107122132426171819 → __106142131416271819__…

__10512223142518__→ 10414213142518 → 10512213341518 →

__10512223142518__…

__51222314251718__→ 41421314251718 → 51221334151718 →

__51222314251718__

But all that is base ten. What about other bases? In fact, nearly all self-descriptive numbers in base ten are also self-descriptive in other bases. An infinite number of other bases, in fact. 22 is a self-descriptive number for all b > 2. The sequence seeded with 1 is identical in all b > 4:

1 → 11 → 21 → 1112 → 3112 → 211213 → 312213 → 212223 → 114213 → 31121314 → 41122314 → 31221324 → __21322314__ → __21322314__…

In bases 2, 3 and 4, the sequence seeded with 1 looks like this:

1 → 11 → 101 → 10101 → 100111 → 1001001 → 1000111 → __1101001__ → __1101001__… (b=2) (1101001_{[2]} = 105 in base 10)

1 → 11 → 21 → 1112 → 10112 → 1010112 → 2011112 → __1011122__ → __1011122__… (b=3) (1011122_{[3]} = 854 in base 10)

1 → 11 → 21 → 1112 → 3112 → 211213 → 312213 → 212223 → 1110213 → 101011213 → 201111213 → __101112213__ → __101112213__… (b=4) (101112213_{[4]} = 71079 in base 10)

In base 2 there are only two self-descriptive numbers (and no loops):

__111__ → three 1s → __111__… (b=2) (111 = 7 in base 10)

__1101001__ → three 0s, four 1s → __1101001__… (b=2) (1101001 = 105 in base 10)

So if you apply the “count digits” procedure in base 2, all numbers, except 111, begin a sequence that ends in 1101001. Base 3 has a few more self-descriptive numbers and also some loops:

__22__ → __22__… (b >= 3)

__10111__ → one 0, four 1s → __10111__… (b=3)

__11112__ → four 1s, one 2 → __11112__…

__100101__ → three 0s, three 1s → __100101__… (b=3)

__1011122__ → one 0, four 1s, two 2s → __1011122__… (b=3)

__2021102__ → two 0s, two 1s, three 2s → __2021102__… (b=3)

__10010122__ → three 0s, three 1s, two 2s → __10010122__…

__2012112__→ 10101102 → 10011112 →

__2012112__…

__10011112__→ 2012112 → 10101102 →

__10011112__…

__10101102__→ 10011112 → 2012112 →

__10101102__…

A question I haven’t been able to answer: Is there a base in which loops can be longer than these?

__103142132415__ → 104122232415 → __103142132415__…

__10512223142518__ → 10414213142518 → 10512213341518 → __10512223142518__…

A question I have been able to answer: What is the sequence when it’s seeded with the title of this blog-post? T4K1NGS3LF13S is a number in all bases >= 30 and its base-30 form equals 15,494,492,743,722,316,018 in base 10 (with the factorization 2 * 72704927 * 106557377767). If T4K1NGS3LF13S seeds a sequence in any b >= 30, the result looks like this:

T4K1NGS3LF13S → 2123141F1G1K1L1N2S1T → 813213141F1G1K1L1N1S1T → A1122314181F1G1K1L1N1S1T → B1221314181A1F1G1K1L1N1S1T → C1221314181A1B1F1G1K1L1N1S1T → D1221314181A1B1C1F1G1K1L1N1S1T → E1221314181A1B1C1D1F1G1K1L1N1S1T → F1221314181A1B1C1D1E1F1G1K1L1N1S1T → G1221314181A1B1C1D1E2F1G1K1L1N1S1T → F1321314181A1B1C1D1E1F2G1K1L1N1S1T → F1222314181A1B1C1D1E2F1G1K1L1N1S1T → __E1421314181A1B1C1D1E2F1G1K1L1N1S1T__ → F1221324181A1B1C1D2E1F1G1K1L1N1S1T → __E1421314181A1B1C1D1E2F1G1K1L1N1S1T__

How do these sequences look if they’re visualized? (Maybe using something like the Ulam Spiral, or one of other visualization techniques you use here)

The OEIS allows you to seem them as graphs:

http://oeis.org/A047841

But there are too few of them to be interesting as a Ulam spiral, I assume.