33373577553133595672223921919239373222222331312236123372233132611...

Take the sequence 3 digits at a time, like so:

333 735 775 531 335 956 722 239 219 192 393 732 ...

Prime factor each 3-digit chunk and put the prime factors in nondecreasing order. For example, 333 = 3 x 3 x 37, 735 = 3 x 5 x 7 x 7 . If a chunk begins with one or more 0's, treat the number as if they weren't there. Values 0 and 1 generate the empty string as output since they have no prime factors (I don't think this occurs in the sequences discussed here). Look at what happens:

Original sequence:

333 735 775 531 335 956 722 239 219 192 393 732 ...

Prime factor sequence:

3x3x37 3x5x7x7 5x5x31 3x3x59 5x67 2x2x239 2x19x19 239 3x73 ...

Note that the sequence of digits in both sequences is the same! I have generated the first 30 million digits of the sequence above and it does not seem to get into a repeating pattern (at least the sequence does not "start over" with the beginning 333735772... anywhere in the first 30 million digits). Rather long blocks of the beginning sequence occur, but they're always followed by something different than at the beginning.

I call this a "self-generating sequence" because using this particular rule and block size, the rule can be applied to the original sequence to recover the digits of the sequence.

For block size 3, there is only one other sequence with the above property using that particular prime factor rule.

It begins:

3337357722193373337333773313293317473313383331213133337111131333...

Grouped in three digits chunks:

333 735 772 219 337 333 733 377 331 329 331 747 ...

Applying the same prime factorization rule gives:

3x3x37 3x5x7x7 2x2x193 3x73 337 3x3x37 733 13x29 331 7x47 ...

Weird, huh? I did web searches looking for these sequences and found nothing. I registered these two sequences with the Online Encyclopedia of Integer Sequences at http://www.research.att.com/~njas/sequences/ .

There are no such sequences for block sizes 1, 2, 4, 5, 6, 8, or 9. That's as far as I've tested. There are 6 of these self-generating sequences for blocksize 7. The beginnings of these sequences are:

3331233 7922224 9513911 8649012 ...
3331233 7922224 9513926 1779832 ...
3331233 7922224 9513926 1779835 3559672 ...
3331233 7922224 9513926 1779835 3559675 ...
3331233 7933329 3827228 7710911 ...
3331233 7933329 3827228 7710912 ...

These are the only other 6 sequences in base 10 that I know of that have this property using the prime factorization rule described above. I've given just enough of each to allow generation of any of them to as many digits as one wishes.

I have found quite a few sequences like the above in bases other than 10. I've also found sequences like the above with similar properties that use different replication rules than the simple prime factoring method I illustrated.

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Evans A Criswell
Huntsville-Decatur Movie Theatre Information Site

It's a way over my head. I keep looking for a punchline.

33373577553133595672223921919239373222222331312236123372233132611...

quote:

---

Evans,
This problem reminded me of the sequence questions on intelligence tests, which I was almost always able to solve,they being far simpler than this one. This had me completely stumped until you explained
the sequence.

---

I don't like those sequence problems on intelligence tests. Here is a simple reason why:

What is the next number in this sequence:

1 2 4 ?

Believe it or not, I can give any answer and justify it.

I can say 7, since from 1 to 2 increases by 1, from 2 to 4 increases by 2, so from 4 to 7 increases by 3, keeping the pattern.

I can say 8, and say each number is the double of the previous number.

I can make up any answer and fit a cubic polynomial (or any other type of function with four parameters) to the four points and say that the function that I fit is the generator.

A more intelligent person may come up with a more complex generator that fits the numbers than a less intelligent person with a more limited set of techniques to try. Just because someone doesn't come up with the particular rule that the test writer used doesn't mean he is wrong or less intelligent.

quote:

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The FIRST thirty MILLION digits!! You do have remarkable persistence.

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It only takes a few minutes to generate the first 30 million digits on recent computers. The result is a 30MB file (actually a 28.61 MB file if you want to be computer sciencically correct) with all the digits in a row.

quote:

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Will you persue it futher, or is that considered proof in mathematics?
Will you present your findings in a professional paper?

---

I am always trying new things. I am in the process of rewriting my hastily-thrown-together C programs that I first used in C++ in a manner to make it easy to plug in new rules. However, mathematical proofs are far more difficult than demonstrating that something works N times or that it works out to N digits for some finite N.

There may be a paper in this somewhere. One professor here wants me to write a paper on it. The day after I found the first sequence I posted above, he wanted me to show it to his Algorithms class he was teaching. We wrote two copies of the sequence all across the board, one beneath the other separated by about a foot, with the digits lining up. We bracketed the top one into 3-digit chunks and the bottom one according to the prime factorization groupings, inserting dots for multiplication signs, and drew lines between the corresponding blocks. It looked awesome.

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Evans A Criswell
Huntsville-Decatur Movie Theatre Information Site

232933115232329747328323411...

Blocked in 3's:

232 933 115 232 329 747 328 323 411 ...

Prime factored:

2^3 x 29, 3 x 311, 5 x 23, 2^3 x 29, 7 x 47, 3^2 x 83, 2^3 * 41, ...

I believe this goes forever, and the sequences of digits in both are the same.

The replication rule is different than the originals I posted. Here, if a prime factor appears multiple times, use exponent notation instead of writing the prime factors multiple times. As before, use nondecreasing order for the prime factors of each number. For example, 100 produces 2^2 x 5^2 rather than 2x2x5x5 as in my first sequences in my original post. It looks a lot cooler if you use superscripts for exponents, which I don't think I can do here in the forum. a^b means a to the b power.

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Evans A Criswell
Huntsville-Decatur Movie Theatre Information Site

I was however able to remember a series of 11 three digit numbers and repeat them in any requested order. That was when I was younger though, I have had way too much brain damage from having children, and I cant even remember who they are from day to day.

quote:

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I will second that opinion Evans on the iq tests. Sequence generators do not themselves denote intelligence, but rather can only denote mathematical savantism. Thats it. I failed miserably on all numerical generators on all my IQ tests in the past, but still managed a 168 on the scale.

---

Yes. Some people may disagree with what I said and say that the most intelligent person will always choose the "simplest" rule that generates the numbers in the sequence. The problem I have with that is how is "simplicity" determined? Let's consider the simple 1, 2, 4, ? sequence. I can say that the rule that the difference between successive numbers goes up by 1 each time is the rule, or that doubling is the rule. Which is simpler? I can say that the first is simpler because it only involves addition, since multiplication is more complex, or I could argue that the doubling rule is simpler using another argument. When taking tests, I've far too often found myself faced with having to guess what the person making the test was thinking because I could thing of more than one correct answer or more than one way to do something.

The nice thing about the sequences I'm coming up with (the self-generating ones) is that

1. I'm specifying a rule and trying to determine how many sequences exist that "self-generate" under the rule.

or

2. I'm trying to find new interesting rules that are simple that have such sequences generated using them.

And most importantly, they're not used for testing. I guess someone could now put a quetion on a test saying "What is the next number in this sequence: 333, 735, 775, ?". Well, I'd guess "531". Someone else might find a different logical relationship between the numbers (and have infinitely many to choose from) and guess another answer.

Steve

 |  IP: Logged

quote:

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And you're NOT employed at Los Almos, or some other neuclear facility??

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Well, I'm not too far off -- I'm employed at the University of Alabama in Huntsville and I do research work that is funded mainly through NASA. When I started, my office was at Marshall Space Flight Center in the Earth Science and Applications Division. Our division was moved off site in 1994, although it is still a NASA facility. Our group is currently spread across two locations right up the street from each other, with about two thirds of us in the NSSTC building (a NASA/UAH building) and the rest of us in Technology Hall (the UAH building). My office has been at Tech Hall since September 1999, and our research center is on the same floor as the computer science department (where I'm working on my Ph.D.).

quote:

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All I could think was man do you have time on your hands. Then again, sequences are almost always named after someone...so here's to the Criswell sequences!

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I was wondering how long it would be before someone said that I had too much time on my hands. After discovering those sequences, I sent an email to a math professor that I'd had for my 3 required graduate level math classes (I chose the combinatorics ones) and asked him what he thought of them and he sent an email back saying that I had too much time on my hands. It's amazing that one of the computer science professors got really excited about the sequences, while that math professor didn't. The CS professor that got excited about them started calling them Criswell sequences the day I showed them to him.

Evans

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Evans A Criswell
Huntsville-Decatur Movie Theatre Information Site

I smell a published paper in here somewhere!

quote:

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Evans, if you don't mind my asking, what led you to these sequences? Are you doing crytographic key or image processing research?

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I have always been a math person. In summer 1976, my father taught me a lot of math. Before then, math didn't really mean anything to me except something I had to do in school to keep from being in trouble. When I started learning more things, I started seeing the real use of everything. I remember measuring oranges to compute their volume and such. I turned 8 that summer.

Enough background...

I've always been a math person since then and have played with math a lot in my spare time. Throught the years, I've played with prime factorizations of numbers and if I'd done things a little differently back in 1989, I would have discovered these sequences then. Back then, I tried the silly process of prime factoring a number, then appending the prime factors in nondecreasing order to make a new number, then prime factoring it. Obviously, I didn't get very far with it since the numbers quickly get too big for computers to represent or prime factor in a timely manner (especially 33 MHz 386's of 1989).

For some strange reason, back in October 2001, I decided to play with that again, and to get around the problem of the number needing prime factoring growing without bound, I decided to treat each sequence of digits in blocks, so that no number larger than the blocksize would get prime factored.

The first night, I simply tried the reproduction rule on some numbers. Here are some examples:

0 --> empty string
1 --> empty string
2 --> 2 (converged)
3 --> 3 (converged)
4 --> 22 --> 211 --> 211 (converged)
5 --> 5 (converged)
6 --> 23 --> 23 (converged)
7 --> 7 (converged)
8 --> 222 --> 2337 --> 2337 (converged)
9 --> 33 --> 311 --> 311 (converged)
10 --> 25 --> 55 --> 511 --> 773 --> 773 (converged)
11 --> 11 (converged)
12 --> 223 --> 223 (converged)
13 --> 13 (converged)
14 --> 27 --> 333 --> 3337 --> 33377 --> 3337711 --> 33373257 ...

Note that 14 is the first number that produces sequences that grow longer with each iteration. It is easy to see that this is the case since "333" produces "3337", which always adds one character to the length of the next sequence. All prime factorization output strings of numbers produce strings at least as long as the number being prime factored, with "0" and "1" being the only exceptions.

Well, I started trying random 3-digit numbers, and found that many converge to a stable string, while others grow without bound, going through cycles of changing prefixes. The first number I tried that produced a self-generating sequence was 135. It's neat to look at it. I'll write each successive string on a different line:

135
3335
33375
3337355
333735775
333735775531
3337357755313359
333735775531335956733

The beginning of the sequence that is staying the same is getting longer. When I first saw this, I found it quite startling, and thought, "What does it mean?" I tried to come up with a visual for it and quickly figured the significance of how successive chunks of 3 digits of the sequence could be paired with successive chunks of the same sequence which associate numbers with their prime factorizations. I'd never seen anything like it, and I did WWW searches to see if anyone else had stumbled upon it.

The next day, I discovered the other sequence. It turns out that the number 460 was the first number I tried that generated a different self-generating sequence: I'll show it as well:

460
22523
335523
567523
33337523
3337355523
333735772223233
3337357722193223233
3337357722193373272317193

Note that the same thing is happening. The beginning of the sequence is "converging" to a stable value and the convergent part is increasing in length.

It turns out that:

The numbers 20, 135, 225, 502, and 980 will generate the sequence beginning with 333735775....

The numbers 460, 620, 700, 820, and 940 will generate the sequence beginning with 333735772....

Using the computer, I found that these were the only two self-generating sequences using block size 3. I wrote a program to look for these sequences directly, without trying to generate them from 3-digits numbers, since it is possible that such self-generating sequences could exist that could not be generated from a single 3-digit number.

In searching for these types of sequences, the first thing to do is find "seeds". I use that term to mean a string that, when replicated by whatever rule is in place, produces itself plus more characters.

Using the prime factorization rule in the original post, here is a list of seeds for different block sizes:

blocksize 1: no seeds
blocksize 2: no seeds
blocksize 3: 333 (produces 3337 [3 x 3 x 37])
blocksize 4: no seeds
blocksize 5: 22564 (produces 225641 [2 x 2 x 5641])
blocksize 6: 210526 (produces 2105263 [2 x 105263])
blocksize 6: 252310 (produces 25231097 [2 x 5 x 23 x 1097])
blocksize 7: 1143241 (produces 11432417 [11 x 43 x 2417])
blocksize 7: 3331233 (produces 333123379 [3 x 3 x 3 x 123379])
blocksize 7: 3710027 (produces 37100271 [37 x 100271])
blocksize 8: no seeds
blocksize 9: 433100023 (produces 4331000231 [433 x 1000231])

That's as far as I've tested looking for seeds using this rule. Unfortunately, the "333" and "3331233" seeds are the only ones that have any associated "self-generating" sequences.