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Thread: A slice of pi

  1. #1
    Join Date
    Jan 2013
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    147

    Smile A slice of pi

    There are some interesting loops in pi digits, thought I'd toss this out there for funsies

    Excuse my novice notation, haha, Richard, if you have a more elegant way to write this out, please do

    Following the rule that : the string becomes the position which becomes the string ( I'll use " Sn " for string number and " Pn " for position number, I started with the numbers 0 through 9, and located the number 0, the position of the number ( after the decimal ) becomes the next string to locate ( Sn → Pn → Sn )

    The process is easy, you start with the number and find it's position in pi digits, the position of the number becomes the next string to search for.

    For 0, 2 through 7, and 9, the series converge on each other fairly quick, 1 is a self-locating digit in pi ( a rarity in itself )

    However, at 8 something neat happens:

    As the series Sn → Pn → Pn progresses for 8, at the 12th step, you come to 3332, and progressing the series, it loops back on itself to 3332 after 11 steps.

    The string 3332 occurs at position 48033, following the rule, it comes back to 3332, ( → ∞ )

    I have not been able to find where and when the numbers actually loop back to themselves, if they do at all, However for the number 8 it seems impossible, following that rule of Sn → Pn → Sn :O

    -----------------

    There are some other loops, here's one:

    The number 169, located at the 40th place, following Sn → Pn → Pn , loops back to 196 after 21 steps in the series

    So, if you like:

    1. Find when and where the numbers 0 - 7, and 9 loop on themselves in pi digits following the rule defined above, or, state why they cannot. ( I so far have not found them to loop )

    2. There are two numbers we cannot do this with, 1 and 8

    Explain why it's impossible with the number 8

    3. Find a proof of whether or not there are an infinite amount of looping numbers in pi

    Here are the numbers 0 through 9, following the rule:

    0 )→ 32 → 15 → 3 → 9 → 5 → 4 → 2 → 6 → 7 → 13...

    1 )→ 1 → ∞

    2 )→ 6 → 7 → 13 → ...

    3 )→ 9 → 5 → 4 → 2 → 6 → 7 → 13...

    4 )→ 2 → 6 → 7 → 13...

    5 )→ 4 → 2 → 6 → 7 → 13...

    6 )→ 7 → 13 → ...

    7 )→ 13 → ...

    8 )→ 11 → 94 → 58 → 10 → 49 → 57 → 404 → 1272 → 8699 → 3292 → 3332 → 48033 → 90311 → 7573 → 1959 → 985 → 2154 → 5276 → 33991 → 18316 → 32928 → 3332 → ∞

    9 )→ 5 → 4 → 2 → 6 → 7 → 13...

    --------

    8 of the series converge at 13
    8 of the series converge at 7
    7 of the series converge at 6
    4 of the series converge on 5
    4 of the series converge on 4
    6 of the series converge on 2

    For 0 and 2 -9 , the series grow fast and and the strings show no signs of shrinking back towards their starting values

    ---------

    Have fun, and if you can simplify the notation or find any more looping numbers, please add them

    Cheers

  2. #2
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    Jun 2007
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    15,146
    Quote Originally Posted by Snakeboy View Post
    There are some interesting loops in pi digits, thought I'd toss this out there for funsies

    Excuse my novice notation, haha, Richard, if you have a more elegant way to write this out, please do

    Following the rule that : the string becomes the position which becomes the string ( I'll use " Sn " for string number and " Pn " for position number, I started with the numbers 0 through 9, and located the number 0, the position of the number ( after the decimal ) becomes the next string to locate ( Sn → Pn → Sn )

    The process is easy, you start with the number and find it's position in pi digits, the position of the number becomes the next string to search for.

    For 0, 2 through 7, and 9, the series converge on each other fairly quick, 1 is a self-locating digit in pi ( a rarity in itself )

    However, at 8 something neat happens:

    As the series Sn → Pn → Pn progresses for 8, at the 12th step, you come to 3332, and progressing the series, it loops back on itself to 3332 after 11 steps.

    The string 3332 occurs at position 48033, following the rule, it comes back to 3332, ( → ∞ )

    I have not been able to find where and when the numbers actually loop back to themselves, if they do at all, However for the number 8 it seems impossible, following that rule of Sn → Pn → Sn :O

    -----------------

    There are some other loops, here's one:

    The number 169, located at the 40th place, following Sn → Pn → Pn , loops back to 196 after 21 steps in the series

    So, if you like:

    1. Find when and where the numbers 0 - 7, and 9 loop on themselves in pi digits following the rule defined above, or, state why they cannot. ( I so far have not found them to loop )

    2. There are two numbers we cannot do this with, 1 and 8

    Explain why it's impossible with the number 8

    3. Find a proof of whether or not there are an infinite amount of looping numbers in pi

    Here are the numbers 0 through 9, following the rule:

    0 )→ 32 → 15 → 3 → 9 → 5 → 4 → 2 → 6 → 7 → 13...

    1 )→ 1 → ∞

    2 )→ 6 → 7 → 13 → ...

    3 )→ 9 → 5 → 4 → 2 → 6 → 7 → 13...

    4 )→ 2 → 6 → 7 → 13...

    5 )→ 4 → 2 → 6 → 7 → 13...

    6 )→ 7 → 13 → ...

    7 )→ 13 → ...

    8 )→ 11 → 94 → 58 → 10 → 49 → 57 → 404 → 1272 → 8699 → 3292 → 3332 → 48033 → 90311 → 7573 → 1959 → 985 → 2154 → 5276 → 33991 → 18316 → 32928 → 3332 → ∞

    9 )→ 5 → 4 → 2 → 6 → 7 → 13...

    --------

    8 of the series converge at 13
    8 of the series converge at 7
    7 of the series converge at 6
    4 of the series converge on 5
    4 of the series converge on 4
    6 of the series converge on 2

    For 0 and 2 -9 , the series grow fast and and the strings show no signs of shrinking back towards their starting values

    ---------

    Have fun, and if you can simplify the notation or find any more looping numbers, please add them

    Cheers
    Hey there Snakeboy,

    It's an interesting coincidence that you bring up loops in pi because I wrote a little program yesterday to confirm your previous observations about pi, and for fun I modified it to look for loops in the first million digits. But your comments do not seem entirely accurate. The numbers don't cycle or converge like you say they do. I have only found a few numbers that cycle even when looking in 200 million digits of pi. A good example of a cycle starts with 37:

    37 => 46 => 19 => 37

    And here is another cycle that begins with the number 40:

    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40

    The number 8 does not cycle in the first 200 million digits of pi. I used this page to check the results that were larger than a million (which is all my little program could handle). Here is how it begins:

    8 => 11 => 94 => 58 => 10 => 49 => 57 => 404 => 1272 => 8699 => 3292 => 3332 => 48033 => 90311 => 112817 => 1149731 => 24909936 => 1838500 => 5264650 => 29232231 => 76236585 => 64535680 => 49758988 => 191873638 => not found

    The "not found" means that it the number 191873638 does not appear in the first 200 million digits of pi. It did not "converge" to anything. It did not cycle. The sequence could continue increasing forever for all we know.

    I think you mistake is to use the word "converge." In mathematics, that word means that a sequence gets closer and closer to a single value. For example, consider the sequence defined by ratios of Fibonacci numbers:

    R(n) = F(n)/F(n-1)

    This sequence converges to the Golden Ratio: R(n) => Phi as n => infinity. It begins 1, 2, 1.5, 1.666..., 1.6, 1.625, etc.

    Now I do understand what you are getting at when you say that they "converge" - what you mean is that two sequences beginning with different starting numbers merge and become the same sequence, whether or not the sequence diverges to infinity or loops. I don't find that particularly significant because the you were just looking at single digits, and they are first found near the beginning so their place numbers would be expected to be small, and hence coincide with small integers.

    In any case, math is fun. Thanks for sharing your research.

    Richard
    • Skepticism is the antiseptic of the mind.
    • Remember why we debate. We have nothing to lose but the errors we hold. Who but a stubborn fool would hold to errors once they have been exposed?

    Check out my blog site

  3. #3
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    Jan 2013
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    147
    Sweet, I figured you'd catch anything, it's good you caught that early, it seemed odd, lol.

    Anyway, I'm up to 20 so far, 16 seems to go to 40 and loop back, and 19 seems to loop on 37

    You ever see how the Fibonacci numbers can all be reduced down to the same 24 digit string ?

    I thought that was pretty cool, read about it while trying to learn some series analysis.

  4. #4
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    Jan 2013
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    147
    aha, I just realized the search window on my pi search page seems to randomly count a space as a digit of some sort

    I'll have to try your page

  5. #5
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    Well, this does seem interesting, excluding my hare-brained mistakes right off the bat

    Is it safe to say that all numbers within a loop are themselves looping numbers ?

    Like the 40 to 40 loop you gave as an example.

  6. #6
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    Jan 2013
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    So I went through the first 50 numbers, double checked everything, so I hope this is correct

    16, 19, 23, 37, 39, 40, 43, 45, 46 will all end up in a loop

    Something neat, I thought, :

    19*37 are factors of the triangular figurate number 703

    Number 19 will loop and have 46, 37, and 19 in the series of the loop

    19 numbers down the line:

    Number 37 loops on itself with 46, 19 and 37 in the series in the loop

    Further down the number line is number 46, which loops on itself with 19, 37 and 46 in the loop

    ---------

    Numbers 16, 23, 39, 40, 43 all have the same loop as well, consisting of 20 numbers, in the series of the loop.

    1, 14, 21, 45 all loop back to the first self-referencing digit in pi, 1

  7. #7
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    Yakima, Wa
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    Quote Originally Posted by Snakeboy View Post
    Well, this does seem interesting, excluding my hare-brained mistakes right off the bat

    Is it safe to say that all numbers within a loop are themselves looping numbers ?

    Like the 40 to 40 loop you gave as an example.
    Yes, any number that is part of or leads to a loop could be called a "looping" numbers. The other numbers would be divergent because they lead to a sequence that ultimately goes to infinity. I doubt there is any way to prove that any numbers ultimately diverge, since that would probably require knowing all the digits of pi. And it seems unlikely that even most numbers loop because the density of the digits smaller than "place number n" keeps getting thinner.

    To understand this, consider how many distinct 2 digit numbers we could fit in the first hundred digits of pi:

    x1 (i.e 11, 21, 31, etc.)
    x2 (i.e. 12, 22, 32, etc.)
    x3 (i.e. 13, 23, 33, etc.)

    The problem is that we are concatenating them and we only have a string of 100 digits to work with.

    x0: 102030405060708090
    x1: 11213141516171819
    x2: 12232425262728292
    x3: etc.

    As you can see, each line requires about 18 digits to cover all the possibilities. But there are ten lines. So that means we need about 180 digits, but we only have 100, so about 45% of the two digit starting numbers won't be found in the first hundred digits, in which case the position number will be larger than the starting number and the sequence will diverge. So I predict that cycles will be pretty rare in the digits of pi, or any sequence of random numbers. With a little work this could be written as a formal theorem.
    • Skepticism is the antiseptic of the mind.
    • Remember why we debate. We have nothing to lose but the errors we hold. Who but a stubborn fool would hold to errors once they have been exposed?

    Check out my blog site

  8. #8
    Join Date
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    Quote Originally Posted by Snakeboy View Post
    So I went through the first 50 numbers, double checked everything, so I hope this is correct

    16, 19, 23, 37, 39, 40, 43, 45, 46 will all end up in a loop

    Something neat, I thought, :

    19*37 are factors of the triangular figurate number 703

    Number 19 will loop and have 46, 37, and 19 in the series of the loop

    19 numbers down the line:

    Number 37 loops on itself with 46, 19 and 37 in the series in the loop

    Further down the number line is number 46, which loops on itself with 19, 37 and 46 in the loop

    ---------

    Numbers 16, 23, 39, 40, 43 all have the same loop as well, consisting of 20 numbers, in the series of the loop.

    1, 14, 21, 45 all loop back to the first self-referencing digit in pi, 1
    So you are doing this all by hand, putting the numbers in one at a time? You should learn a little javascript programming. It does all the work for you. Here are all the looping numbers less than a thousand found in the first million digits of pi. (There may be more if we used a larger set of pi digits.) It took about ten minutes to write, and 5 seconds to run.

    ----------------
    Start: 1
    1
    ----------------
    Start: 14
    1
    1
    ----------------
    Start: 16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 19
    37
    46
    19
    ----------------
    Start: 21
    93
    14
    1
    1
    ----------------
    Start: 23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 37
    46
    19
    37
    ----------------
    Start: 39
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 45
    60
    127
    297
    737
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 46
    19
    37
    46
    ----------------
    Start: 60
    127
    297
    737
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 61
    219
    716
    39
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    70
    ----------------
    Start: 71
    39
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 73
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 93
    14
    1
    1
    ----------------
    Start: 96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    70
    96
    ----------------
    Start: 115
    921
    422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 127
    297
    737
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 141
    1
    1
    ----------------
    Start: 165
    238
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 169
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    ----------------
    Start: 180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    70
    96
    180
    ----------------
    Start: 183
    490
    907
    542
    700
    306
    115
    921
    422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 197
    37
    46
    19
    37
    ----------------
    Start: 211
    93
    14
    1
    1
    ----------------
    Start: 219
    716
    39
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 226
    964
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    70
    96
    180
    ----------------
    Start: 238
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 264
    21
    93
    14
    1
    1
    ----------------
    Start: 270
    165
    238
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 274
    462
    19
    37
    46
    19
    ----------------
    Start: 286
    73
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 297
    737
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 306
    115
    921
    422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 326
    274
    462
    19
    37
    46
    19
    ----------------
    Start: 333
    1698
    25318
    33479
    45671
    88095
    40332
    4592
    60
    127
    297
    737
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 356
    615
    1029
    8196
    197
    37
    46
    19
    37
    ----------------
    Start: 375
    46
    19
    37
    46
    ----------------
    Start: 394
    526
    612
    219
    716
    39
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 399
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 406
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    70
    ----------------
    Start: 422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 433
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 440
    511
    394
    526
    612
    219
    716
    39
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 459
    60
    127
    297
    737
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 462
    19
    37
    46
    19
    ----------------
    Start: 464
    1159
    921
    422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 490
    907
    542
    700
    306
    115
    921
    422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 495
    464
    1159
    921
    422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 511
    394
    526
    612
    219
    716
    39
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 514
    2117
    93
    14
    1
    1
    ----------------
    Start: 526
    612
    219
    716
    39
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 542
    700
    306
    115
    921
    422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 545
    4338
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 557
    1101
    2779
    15008
    15375
    202762
    107788
    780706
    96071
    53594
    141
    1
    1
    ----------------
    Start: 607
    286
    73
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 609
    127
    297
    737
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 612
    219
    716
    39
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 615
    1029
    8196
    197
    37
    46
    19
    37
    ----------------
    Start: 648
    226
    964
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    70
    96
    180
    ----------------
    Start: 656
    514
    2117
    93
    14
    1
    1
    ----------------
    Start: 665
    211
    93
    14
    1
    1
    ----------------
    Start: 700
    306
    115
    921
    422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 706
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    70
    96
    ----------------
    Start: 714
    609
    127
    297
    737
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 716
    39
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 737
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 806
    967
    1400
    4365
    11353
    127494
    4220
    2371
    1925
    1166
    3993
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 819
    197
    37
    46
    19
    37
    ----------------
    Start: 872
    648
    226
    964
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    70
    96
    180
    ----------------
    Start: 874
    1949
    495
    464
    1159
    921
    422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 902
    714
    609
    127
    297
    737
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 903
    356
    615
    1029
    8196
    197
    37
    46
    19
    37
    ----------------
    Start: 907
    542
    700
    306
    115
    921
    422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 921
    422
    1839
    9323
    14
    1
    1
    ----------------
    Start: 931
    440
    511
    394
    526
    612
    219
    716
    39
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 932
    14
    1
    1
    ----------------
    Start: 937
    45
    60
    127
    297
    737
    299
    2643
    21
    93
    14
    1
    1
    ----------------
    Start: 943
    399
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 964
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    70
    96
    180
    ----------------
    Start: 967
    1400
    4365
    11353
    127494
    4220
    2371
    1925
    1166
    3993
    43
    23
    16
    40
    70
    96
    180
    3664
    24717
    15492
    84198
    65489
    3725
    16974
    41702
    3788
    5757
    1958
    14609
    62892
    44745
    9385
    169
    40
    ----------------
    Start: 996
    459
    60
    127
    297
    737
    299
    2643
    21
    93
    14
    1
    1
    • Skepticism is the antiseptic of the mind.
    • Remember why we debate. We have nothing to lose but the errors we hold. Who but a stubborn fool would hold to errors once they have been exposed?

    Check out my blog site

  9. #9
    Join Date
    Jan 2013
    Posts
    147
    Impressive, Richard.

    I have been doing this by hand, unfortunately, also cross-referencing them against some other things, so it's rather tedious.

    I figured for a really complex analysis it would be a no-brainer to have a program do the looking, I unfortunately have not programmed since learning Basic on the Apple IIe



    I've not been able to find too much work in this area, aside from a little bit on the OEIS

    http://oeis.org/A232013

    Apparently these are called " orbits ", seems fitting.

  10. #10
    Join Date
    Jan 2013
    Posts
    147
    Happy Pi day !

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