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Patterns in the b’-numbers related to TM #4 (chaotic) presumptive sequence

Patterns in the b’-numbers related to TM #4 (chaotic) presumptive sequence

 

b’ numbers =

((( {1, 2, 3, 4, 5, 6, 7}

Union the following disjoint sets,
one per number of bits in the base 2
representations of these numbers,
4 bits for v8,
5 bits for v16,
6 bits for v32,
7 bits for v64,
8 bits for v128,
9 bits for v256,
10 bits for v512,
11 bits for v1024
and so on.
)))

What I noticed is that
if we subtract 128 from the 16 numbers in v128, then we get
the first 16 numbers of “v256 minus 256”, which has 23 numbers.

In other words, the offsets from 16 in [17, 18, 19, 22, 26]
give the 5 first terms in the offsets from 32 in [33, 34, 35, 38, 42, 50],
and so on for higher powers of 2.

The v8 through v1024 finite sequences :

 

? v8
%225 = [9, 10, 11, 14]

 

? v16
%226 = [17, 18, 19, 22, 26]

 

? v32
%227 = [33, 34, 35, 38, 42, 50]

 

? v64
%228 = [65, 66, 67, 70, 74, 82, 97, 99, 102, 106]

 

? v128
%229 = [129, 130, 131, 134, 138, 146, 161, 163, 166, 170, 193, 195, 198, 202, 227, 234]

 

? v256
%230 = [257, 258, 259, 262, 266, 274, 289, 291, 294, 298, 321, 323, 326, 330, 355, 362, 386, 402, 417, 422, 451, 458, 483]

 

? v512
%231 = [513, 514, 515, 518, 522, 530, 545, 547, 550, 554, 577, 579, 582, 586, 611, 618, 642, 658, 673, 678, 707, 714, 739, 769, 771, 774, 778, 803, 810, 833, 838, 874, 934, 970, 995]

 

? v1024
%232 = [1025, 1026, 1027, 1030, 1034, 1042, 1057, 1059, 1062, 1066, 1089, 1091, 1094, 1098, 1123, 1130, 1154, 1170, 1185, 1190, 1219, 1226, 1251, 1281, 1283, 1286, 1290, 1315, 1322, 1345, 1350, 1386, 1446, 1482, 1507, 1538, 1554, 1569, 1574, 1603, 1610, 1635, 1666, 1682, 1697, 1731, 1795, 1802, 1827, 1862, 1958, 1994]

 

===============

Checking the property:

? for(X=1, c8, print(v8[X]-8))
1
2
3
6
?
? for(X=1, c16, print(v16[X]-16))
1
2
3
6
10
?
? for(X=1, c32, print(v32[X]-32))
1
2
3
6
10
18
?
? for(X=1, c64, print(v64[X]-64))
1
2
3
6
10
18
33
35
38
42
?
? for(X=1, c128, print(v128[X]-128))
1
2
3
6
10
18
33
35
38
42
65
67
70
74
99
106
?
? for(X=1, c256, print(v256[X]-256))
1
2
3
6
10
18
33
35
38
42
65
67
70
74
99
106
130
146
161
166
195
202
227
?
? for(X=1, c512, print(v512[X]-512))
1
2
3
6
10
18
33
35
38
42
65
67
70
74
99
106
130
146
161
166
195
202
227
257
259
262
266
291
298
321
326
362
422
458
483
?
? for(X=1, c1024, print(v1024[X]-1024))
1
2
3
6
10
18
33
35
38
42
65
67
70
74
99
106
130
146
161
166
195
202
227
257
259
262
266
291
298
321
326
362
422
458
483
514
530
545
550
579
586
611
642
658
673
707
771
778
803
838
934
970

==================================

A sequence of offsets:

{1, 2, 3, 6, 10, 18, 33, 35, 38, 42, 65, 67, 70, 74, 99, 106}

would be the first 16 terms in a hypothetically infinite
sequence of offsets.

Sloane’s Online Encyclopedia of Integer Sequences has nothing
for:

1, 2, 3, 6, 10, 18, 33

“Sorry, but the terms do not match anything in the table.”
is the reply.

Can we crack the sequence of offsets?

I’m not so sure.

 

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Written by meditationatae

September 20, 2017 at 3:23 am

Posted in History

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