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The sequence of “b-numbers” determines some values in Column 3 of the TM #4 presumptive sequence

For a b-number ‘b’ ,

empirically,

s(2^m – b – 2) = 2^m – b, as long as
b > 0 (always true) AND
b < 2^(m-1).

Being connected to Column 3 of 8,
a “b-number” b is such that b == 3 (mod 8).

A b-number ‘b’ predicts relationships
of the type:
s(y) = y+2, where y == 3 (mod 8).
So, y+2 == 5 (mod 8), and
2^m – b = y+2 == 5 (mod 8).

Suppose m >= 3, which I think is
always the case, then 2^m == 0 (mod 8),
so -b == 5 (mod 8), so
b == -5 (mod 8) or
b == 3 (mod 8).

 

For m = 3, 2^m = 8. If b = 3, then
3 < 2^(3-1).

It predicts:

s(8-3-2) = 8-3 i.e.
s(3) = 5.

s sequence is: 3, 1, 5, etc. so s(3) = 5 check Ok.

The s-sequence comes from the run-length encoding
of ‘0’ and ‘1’ symbols at special times, in the
evolution of TM #4, “chaotic”, of Heiner Marxen
and Buntrock, found around 1990, and that exhibits
a seemingly chaotic or random behavior.

 

 

For generation number “x”,
the “b-number” ‘b’ has an ancestor
from an earlier generation that is obtained by
subtracting 2^(x-1) from ‘b’.

Example:

b = 555
generation = x = 10

rule: 2^(x-1) = 2^9 = 512
and b – 2^(x-1) =
= 555 – 512 = 43.

43 is the ancestor of 555.

Is 43 in the sequence of b-numbers ?

Yes, from generation 6.

 

Below, b-numbers below 2^14 = 16384:

 

b-number, generation, “ancestor”
===================================
3 2 1

 

11 4 3

 

19 5 3
27 5 11

 

35 6 3
43 6 11
51 6 19

 

67 7 3
75 7 11
83 7 19
107 7 43

 

131 8 3
139 8 11
147 8 19
171 8 43
203 8 75

 

259 9 3
267 9 11
275 9 19
299 9 43
331 9 75
395 9 139

 

515 10 3
523 10 11
531 10 19
555 10 43
587 10 75
651 10 139
771 10 259
787 10 275
811 10 299
843 10 331

 

1027 11 3
1035 11 11
1043 11 19
1067 11 43
1099 11 75
1163 11 139
1283 11 259
1299 11 275
1323 11 299
1355 11 331
1539 11 515
1555 11 531
1579 11 555
1611 11 587
1811 11 787
1867 11 843

 

2051 12 3
2059 12 11
2067 12 19
2091 12 43
2123 12 75
2187 12 139
2307 12 259
2323 12 275
2347 12 299
2379 12 331
2563 12 515
2579 12 531
2603 12 555
2635 12 587
2835 12 787
2891 12 843
3083 12 1035
3211 12 1163
3331 12 1283
3371 12 1323
3603 12 1555
3659 12 1611
3859 12 1811

 

4099 13 3
4107 13 11
4115 13 19
4139 13 43
4171 13 75
4235 13 139
4355 13 259
4371 13 275
4395 13 299
4427 13 331
4611 13 515
4627 13 531
4651 13 555
4683 13 587
4883 13 787
4939 13 843
5131 13 1035
5259 13 1163
5379 13 1283
5419 13 1323
5651 13 1555
5707 13 1611
5907 13 1811
6147 13 2051
6163 13 2067
6187 13 2091
6219 13 2123
6419 13 2323
6475 13 2379
6659 13 2563
6699 13 2603
6987 13 2891
7467 13 3371
7755 13 3659
7955 13 3859

 

8195 14 3
8203 14 11
8211 14 19
8235 14 43
8267 14 75
8331 14 139
8451 14 259
8467 14 275
8491 14 299
8523 14 331
8707 14 515
8723 14 531
8747 14 555
8779 14 587
8979 14 787
9035 14 843
9227 14 1035
9355 14 1163
9475 14 1283
9515 14 1323
9747 14 1555
9803 14 1611
10003 14 1811
10243 14 2051
10259 14 2067
10283 14 2091
10315 14 2123
10515 14 2323
10571 14 2379
10755 14 2563
10795 14 2603
11083 14 2891
11563 14 3371
11851 14 3659
12051 14 3859
12299 14 4107
12427 14 4235
12547 14 4355
12587 14 4395
12819 14 4627
12875 14 4683
13075 14 4883
13323 14 5131
13451 14 5259
13571 14 5379
13843 14 5651
14355 14 6163
14411 14 6219
14611 14 6419
14891 14 6699
15659 14 7467
15947 14 7755

 

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Written by meditationatae

September 18, 2017 at 6:25 pm

Posted in History

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