IWABUCHI Yu(u)ki
November 07, 2021
45

# collatz-bion-experiment-2018-10-31

2018年10月31日のコラッツの問題への取り組み

collatz-bion-experiment-2018-10-31.nb

## IWABUCHI Yu(u)ki

November 07, 2021

## Transcript

1. collatz-bion-experiment-2018-10-31.nb
created at 2018-10-31
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collatz-bion-experiment-2018-10-31.nb

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-0.411069, -0.409763, 0.588468, -0.410376, 0.588062, -0.412975, -0.412292,
0.58679, -0.413821, 0.585773, -0.413957, 0.587839, -0.413123, -0.413762,
0.586661, -0.413906, 0.586469, -0.414034, -0.414369, 0.585408, -0.41474,
0.58516, -0.41451, -0.414686, 0.585197, -0.414881, 0.585796, -0.413927,
0.586441, -0.407175, -0.394279, 0.598637, -0.405992, -0.321928}
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collatz-bion-experiment-2018-10-31.nb
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-0.415019, 0.585011, -0.415005, -0.415016, 0.584977, -0.415028, 0.584975,
-0.415029, 0.584968, -0.41503, -0.415032, 0.584969, -0.415033, 0.584968,
-0.41503, -0.415032, 0.58497, -0.415028, 0.584969, -0.415033, 0.584974,
-0.41503, -0.415033, 0.584966, -0.415035, 0.584964, -0.415034, -0.415016,
0.585019, -0.414888, 0.585062, -0.414971, 0.585007, -0.414801, -0.41488,
0.585382, -0.414478, 0.585336, -0.41454, -0.414706, 0.585183, -0.414743,
0.585355, -0.414514, -0.414689, 0.585195, -0.410088, 0.591535, -0.40632,
0.590745, -0.411195, -0.404842, 0.59172, -0.41055, 0.587946, -0.411069,
-0.409763, 0.588468, -0.410376, 0.588062, -0.412975, -0.412292,
0.58679, -0.413821, 0.585773, -0.413957, 0.587839, -0.413123, -0.413762,
0.586661, -0.413906, 0.586469, -0.414034, -0.414369, 0.585408, -0.41474,
0.58516, -0.41451, -0.414686, 0.585197, -0.414881, 0.585796, -0.413927,
0.586441, -0.407175, -0.394279, 0.598637, -0.405992, -0.321928}
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collatz-bion-experiment-2018-10-31.nb

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196 587 028 480, 147 442 368 512, 110 582 824 960, 82 938 167 296, 62 205 722 624,
46 655 340 544, 34 992 029 696, 26 244 284 416, 19 683 344 384, 14 762 639 360,
11 072 045 056, 8 304 066 560, 6 228 082 688, 4 671 078 400, 3 503 325 184, 2 627 502 080,
1 970 634 752, 1 477 984 256, 1 108 492 288, 831 373 312, 623 534 080, 467 652 608,
350 740 480, 263 057 408, 197 294 080, 147 971 072, 110 978 560, 83 234 048,
62 425 600, 46 819 328, 35 115 008, 26 337 280, 19 755 008, 14 817 280, 11 113 472,
8 335 360, 6 252 544, 4 689 920, 3 518 464, 2 639 872, 1 980 416, 1 485 824, 1 114 624,
836 096, 627 200, 470 528, 353 024, 264 832, 198 656, 149 504, 112 640, 84 992,
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NestWhileList[c2, n, # ≠ 1 &]], 1]},
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,

True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]], {n, 1, 20, 2}] //
���
TableForm
�����������������
1
1 5
4
3
2
1 5
4
1 5
4
13
8
17
8
11
4
7
2
1 5
4
13
8
17
8
11
4
7
2
9
2
1 5
4
13
8
17
8
11
4
1 5
4
13
8
1 5
4
53
32
35
16
23
8
15
4
1 5
4
13
8
17
8
1 5
4
13
8
17
8
11
4
29
8
19
4
������
��
First /@
��
Select[
������
Table[
n →
��
Count[
���������������
NestWhileList[c, n, # ≠ 1 &], _?
������
OddQ], {n, 1, 250, 2}],
��
Last[#] ⩵ 3 &]
������ {3, 13, 53, 113, 213, 227}
16���
collatz-bion-experiment-2018-10-31.nb

17. (* ������������������������ *)
���
Union[
���
Flatten[
�⋯
Table[
��
Reverse[2^
����
With[{li =
��
Mod[
�⋯
Log2[
���������������
NestWhileList[c2, n, # ≠ 1 &]], 1]},
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,

True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]], {n, 1, 200, 2}]]]
% //
��
Numerator
������ 1,
5
4
,
21
16
,
85
64
,
341
256
,
3
2
,
13
8
,
53
32
,
113
64
,
227
128
,
17
8
,
69
32
,
35
16
,
141
64
,
75
32
,
151
64
,
11
4
,
45
16
,
181
64
,
23
8
,
93
32
,
7
2
,
29
8
,
117
32
,
15
4
,
61
16
,
245
64
,
9
2
,
37
8
,
149
32
,
19
4
,
77
16
,
81
16
,
325
64
,
163
32
,
49
8
,
197
32
,
99
16
,
25
4
,
101
16
,
51
8
,
205
32
,
433
64
,
65
8
,
131
16
,
33
4
,
133
16
,
67
8
,
269
32
,
1093
128
,
577
64
,
43
4
,
173
16
,
87
8
,
177
16
,
89
8
,
179
16
,
1457
128
,
3077
256
,
57
4
,
229
16
,
115
8
,
59
4
,
119
8
,
971
64
,
2051
128
,
305
16
,
153
8
,
39
2
,
157
8
,
79
4
,
317
16
,
647
32
,
1367
64
,
203
8
,
209
8
,
105
4
,
211
8
,
431
16
,
911
32
,
135
4
,
139
4
,
281
8
,
287
8
,
607
16
,
2429
64
,
185
4
,
187
4
,
191
4
,
809
16
,
1619
32
,
123
2
,
127
2
,
539
8
,
1079
16
,
169
2
,
359
4
,
719
8
,
239
2
,
479
4
, 159,
319
2
,
425
2
, 283, 377,
502, 668, 888, 890, 1186, 1580, 2104, 2800, 3728, 4960, 6592, 6608, 8768, 8800,
11 648, 11 712, 15 488, 15 520, 20 608, 20 672, 27 392, 27 520, 36 352, 36 608,
48 128, 48 384, 48 640, 63 488, 64 000, 64 512, 64 768, 83 968, 84 480, 84 992,
86 272, 110 592, 111 616, 112 640, 113 152, 114 944, 148 480, 149 504, 150 016,
150 528, 153 088, 197 632, 198 656, 199 680, 203 776, 263 168, 264 192, 350 208
������ {1, 5, 21, 85, 341, 3, 13, 53, 113, 227, 17, 69, 35, 141, 75, 151, 11, 45, 181, 23, 93,
7, 29, 117, 15, 61, 245, 9, 37, 149, 19, 77, 81, 325, 163, 49, 197, 99, 25, 101,
51, 205, 433, 65, 131, 33, 133, 67, 269, 1093, 577, 43, 173, 87, 177, 89, 179,
1457, 3077, 57, 229, 115, 59, 119, 971, 2051, 305, 153, 39, 157, 79, 317, 647,
1367, 203, 209, 105, 211, 431, 911, 135, 139, 281, 287, 607, 2429, 185, 187,
191, 809, 1619, 123, 127, 539, 1079, 169, 359, 719, 239, 479, 159, 319, 425,
283, 377, 502, 668, 888, 890, 1186, 1580, 2104, 2800, 3728, 4960, 6592, 6608,
8768, 8800, 11 648, 11 712, 15 488, 15 520, 20 608, 20 672, 27 392, 27 520, 36 352,
36 608, 48 128, 48 384, 48 640, 63 488, 64 000, 64 512, 64 768, 83 968, 84 480, 84 992,
86 272, 110 592, 111 616, 112 640, 113 152, 114 944, 148 480, 149 504, 150 016,
150 528, 153 088, 197 632, 198 656, 199 680, 203 776, 263 168, 264 192, 350 208}
������ {5, 21, 85, 341}
������ {5, 21, 85, 341}
collatz-bion-experiment-2018-10-31.nb
���17

18. ������ {3, 13, 53, 113, 227}
�������
������ {3, 13, 53, 113, 227}
������
https://oeis.org/A198584
������
������
Table[n →
��
Count[
���������������
NestWhileList[c, n, # ≠ 1 &], _?
������
OddQ], {n, 1, 250, 2}]
��
First /@
��
Select[%,
��
Last[#] ⩵ 3 &]
������ {1 → 1, 3 → 3, 5 → 2, 7 → 6, 9 → 7, 11 → 5, 13 → 3, 15 → 6, 17 → 4, 19 → 7, 21 → 2, 23 → 5,
25 → 8, 27 → 42, 29 → 6, 31 → 40, 33 → 9, 35 → 4, 37 → 7, 39 → 12, 41 → 41, 43 → 10,
45 → 5, 47 → 39, 49 → 8, 51 → 8, 53 → 3, 55 → 42, 57 → 11, 59 → 11, 61 → 6, 63 → 40,
65 → 9, 67 → 9, 69 → 4, 71 → 38, 73 → 43, 75 → 4, 77 → 7, 79 → 12, 81 → 7, 83 → 41,
85 → 2, 87 → 10, 89 → 10, 91 → 34, 93 → 5, 95 → 39, 97 → 44, 99 → 8, 101 → 8, 103 → 32,
105 → 13, 107 → 37, 109 → 42, 111 → 25, 113 → 3, 115 → 11, 117 → 6, 119 → 11,
121 → 35, 123 → 16, 125 → 40, 127 → 16, 129 → 45, 131 → 9, 133 → 9, 135 → 14,
137 → 33, 139 → 14, 141 → 4, 143 → 38, 145 → 43, 147 → 43, 149 → 7, 151 → 4,
153 → 12, 155 → 31, 157 → 12, 159 → 19, 161 → 36, 163 → 7, 165 → 41, 167 → 24,
169 → 17, 171 → 46, 173 → 10, 175 → 29, 177 → 10, 179 → 10, 181 → 5, 183 → 34,
185 → 15, 187 → 15, 189 → 39, 191 → 15, 193 → 44, 195 → 44, 197 → 8, 199 → 44,
201 → 5, 203 → 13, 205 → 8, 207 → 32, 209 → 13, 211 → 13, 213 → 3, 215 → 37, 217 → 8,
219 → 18, 221 → 42, 223 → 25, 225 → 18, 227 → 3, 229 → 11, 231 → 47, 233 → 30,
235 → 47, 237 → 11, 239 → 18, 241 → 6, 243 → 35, 245 → 6, 247 → 16, 249 → 16}
������ {3, 13, 53, 113, 213, 227}
������
������
Table[n →
��
Length[
���������������
NestWhileList[c2, n, # ≠ 1 &]], {n, 1, 250, 2}];
��
First /@
��
Select[%,
��
Last[#] ⩵ 3 &]
������ {3, 13, 53, 113, 213, 227}
������
������
Table[n →
��
Length[
���������������
NestWhileList[c2, n, # ≠ 1 &]], {n, 1, 250, 2}];
��
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��
Select[%,
��
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������ {1}
������
������
Table[n →
��
Length[
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NestWhileList[c2, n, # ≠ 1 &]], {n, 1, 350, 2}];
��
First /@
��
Select[%,
��
Last[#] ⩵ 2 &]
������ {5, 21, 85, 341}
������
������
Table[n →
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Length[
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NestWhileList[c2, n, # ≠ 1 &]], {n, 1, 250, 2}];
��
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��
Select[%,
��
Last[#] ⩵ 10 &]
������ {43, 87, 89, 173, 177, 179}
18���
collatz-bion-experiment-2018-10-31.nb

19. (* ����������� *)
��⋯
SortBy[
���
Union[
���
Flatten[
�⋯
Table[
����
With[{processLi =
���������������
NestWhileList[c2, n, # ≠ 1 &]},
����
With[{valueLi = 2^
����
With[{li =
��
Mod[
�����
Log2[processLi], 1]},
li -
��
Append[
��
Reverse[
��
Accumulate[
��
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True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]},
����
Thread[processLi → valueLi]]], {n, 1, 20, 2}]]],
��
First]
������ 1 → 1, 3 →
3
2
, 5 →
5
4
, 7 →
7
2
, 9 →
9
2
, 11 →
11
4
, 13 →
13
8
,
15 →
15
4
, 17 →
17
8
, 19 →
19
4
, 23 →
23
8
, 29 →
29
8
, 35 →
35
16
, 53 →
53
32

��
Rule
������ (* ���������������� *)
��⋯
SortBy[
���
Union[
���
Flatten[
�⋯
Table[
����
With[{processLi =
���������������
NestWhileList[c2, n, # ≠ 1 &]},
����
With[{valueLi =
��
Reverse[2^
����
With[{li =
��
Mod[
�����
Log2[processLi], 1]},
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,

True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]]},
# →
��
First[
�����
FirstPosition[valueLi, #]] & /@ valueLi]], {n,
1, 200, 2}]]],
��
First]
��
Show[{
����
Plot[2
�����
Log2[3 x - 2], {x, 1, 20},
������
PlotRange → {{1, 20}, {1, 15}}],
����⋯
ListPlot[{
��
First[#],
��
Last[#]} & /@ %,
������
PlotRange → {{1, 20}, {1, 15}}]}]
��
First /@
%%
collatz-bion-experiment-2018-10-31.nb
���19

20. ������ 1 → 1,
5
4
→ 2,
21
16
→ 2,
85
64
→ 2,
341
256
→ 2,
3
2
→ 3,
13
8
→ 3,
53
32
→ 3,
113
64
→ 3,
227
128
→ 3,
17
8
→ 4,
69
32
→ 4,
35
16
→ 4,
141
64
→ 4,
75
32
→ 4,
151
64
→ 4,
11
4
→ 5,
45
16
→ 5,
181
64
→ 5,
23
8
→ 5,
93
32
→ 5,
7
2
→ 6,
29
8
→ 6,
117
32
→ 6,
15
4
→ 6,
61
16
→ 6,
245
64
→ 6,
9
2
→ 7,
37
8
→ 7,
149
32
→ 7,
19
4
→ 7,
77
16
→ 7,
81
16
→ 7,
325
64
→ 7,
163
32
→ 7,
49
8
→ 8,
197
32
→ 8,
99
16
→ 8,
25
4
→ 8,
101
16
→ 8,
51
8
→ 8,
205
32
→ 8,
433
64
→ 8,
65
8
→ 9,
131
16
→ 9,
33
4
→ 9,
133
16
→ 9,
67
8
→ 9,
269
32
→ 9,
1093
128
→ 9,
577
64
→ 9,
43
4
→ 10,
173
16
→ 10,
87
8
→ 10,
177
16
→ 10,
89
8
→ 10,
179
16
→ 10,
1457
128
→ 10,
3077
256
→ 10,
57
4
→ 11,
229
16
→ 11,
115
8
→ 11,
59
4
→ 11,
119
8
→ 11,
971
64
→ 11,
2051
128
→ 11,
305
16
→ 12,
153
8
→ 12,
39
2
→ 12,
157
8
→ 12,
79
4
→ 12,
317
16
→ 12,
647
32
→ 12,
1367
64
→ 12,
203
8
→ 13,
209
8
→ 13,
105
4
→ 13,
211
8
→ 13,
431
16
→ 13,
911
32
→ 13,
135
4
→ 14,
139
4
→ 14,
281
8
→ 14,
287
8
→ 14,
607
16
→ 14,
2429
64
→ 14,
185
4
→ 15,
187
4
→ 15,
191
4
→ 15,
809
16
→ 15,
1619
32
→ 15,
123
2
→ 16,
127
2
→ 16,
539
8
→ 16,
1079
16
→ 16,
169
2
→ 17,
359
4
→ 17,
719
8
→ 17,
239
2
→ 18,
479
4
→ 18, 159 → 19,
319
2
→ 19,
425
2
→ 20, 283 → 21, 377 → 22, 502 → 23, 668 → 24,
888 → 25, 890 → 25, 1186 → 26, 1580 → 27, 2104 → 28, 2800 → 29, 3728 → 30,
4960 → 31, 6592 → 32, 6608 → 32, 8768 → 33, 8800 → 33, 11 648 → 34, 11 712 → 34,
15 488 → 35, 15 520 → 35, 20 608 → 36, 20 672 → 36, 27 392 → 37, 27 520 → 37,
36 352 → 38, 36 608 → 38, 48 128 → 39, 48 384 → 39, 48 640 → 39, 63 488 → 40,
64 000 → 40, 64 512 → 40, 64 768 → 40, 83 968 → 41, 84 480 → 41, 84 992 → 41,
86 272 → 41, 110 592 → 42, 111 616 → 42, 112 640 → 42, 113 152 → 42, 114 944 → 42,
148 480 → 43, 149 504 → 43, 150 016 → 43, 150 528 → 43, 153 088 → 43, 197 632 → 44,
198 656 → 44, 199 680 → 44, 203 776 → 44, 263 168 → 45, 264 192 → 45, 350 208 → 46
������
5 10 15 20
2
4
6
8
10
12
14
20���
collatz-bion-experiment-2018-10-31.nb

21. ������ 1,
5
4
,
21
16
,
85
64
,
341
256
,
3
2
,
13
8
,
53
32
,
113
64
,
227
128
,
17
8
,
69
32
,
35
16
,
141
64
,
75
32
,
151
64
,
11
4
,
45
16
,
181
64
,
23
8
,
93
32
,
7
2
,
29
8
,
117
32
,
15
4
,
61
16
,
245
64
,
9
2
,
37
8
,
149
32
,
19
4
,
77
16
,
81
16
,
325
64
,
163
32
,
49
8
,
197
32
,
99
16
,
25
4
,
101
16
,
51
8
,
205
32
,
433
64
,
65
8
,
131
16
,
33
4
,
133
16
,
67
8
,
269
32
,
1093
128
,
577
64
,
43
4
,
173
16
,
87
8
,
177
16
,
89
8
,
179
16
,
1457
128
,
3077
256
,
57
4
,
229
16
,
115
8
,
59
4
,
119
8
,
971
64
,
2051
128
,
305
16
,
153
8
,
39
2
,
157
8
,
79
4
,
317
16
,
647
32
,
1367
64
,
203
8
,
209
8
,
105
4
,
211
8
,
431
16
,
911
32
,
135
4
,
139
4
,
281
8
,
287
8
,
607
16
,
2429
64
,
185
4
,
187
4
,
191
4
,
809
16
,
1619
32
,
123
2
,
127
2
,
539
8
,
1079
16
,
169
2
,
359
4
,
719
8
,
239
2
,
479
4
, 159,
319
2
,
425
2
, 283, 377,
502, 668, 888, 890, 1186, 1580, 2104, 2800, 3728, 4960, 6592, 6608, 8768, 8800,
11 648, 11 712, 15 488, 15 520, 20 608, 20 672, 27 392, 27 520, 36 352, 36 608,
48 128, 48 384, 48 640, 63 488, 64 000, 64 512, 64 768, 83 968, 84 480, 84 992,
86 272, 110 592, 111 616, 112 640, 113 152, 114 944, 148 480, 149 504, 150 016,
150 528, 153 088, 197 632, 198 656, 199 680, 203 776, 263 168, 264 192, 350 208
(* 12�n�������������n *)
exponent[x_] :=
����
With[{li =
��
Select[
������
FactorInteger[x],
��
First[#] ⩵ 2 &]},
���
If[
��
Length[li] ⩵ 0, 0,
��
Last[
��
Last[li]]]]
������
exponent[12]
������
2
������
exponent[13]
������
0
������
exponent
5
24

������ -3
������ (* ���������������������2������� *)
�������
SortBy
#
2^exponent[#]
, exponent[#] & /@
���
Union[
���
Flatten[
�⋯
Table[
��
Reverse[2^
����
With[{li =
��
Mod[
�⋯
Log2[
���������������
NestWhileList[c2, n, # ≠ 1 &]], 1]},
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,

True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]], {n, 1, 20, 2}]]],
��
First
������ {{1, 0}, {3, -1}, {5, -2}, {7, -1}, {9, -1}, {11, -2}, {13, -3},
{15, -2}, {17, -3}, {19, -2}, {23, -3}, {29, -3}, {35, -4}, {53, -5}}
collatz-bion-experiment-2018-10-31.nb
���21

22. ������ (* ����������������� *)
�������
SortBy
#
2^exponent[#]
, exponent[#] & /@
���
Union[
���
Flatten[
�⋯
Table[
��
Reverse[2^
����
With[{li =
��
Mod[
�⋯
Log2[
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NestWhileList[c2, n, # ≠ 1 &]], 1]},
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,

True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]], {n, 1, 1000, 2}]]],
��
First;
�������
ListPlot[%,
������
PlotRange →
���
All]
�������
ListPlot[%%,
������
PlotRange → {{1, 1000},
���
All}]
������
20000 40000 60000 80000
-10
-5
5
10
15
������
200 400 600 800 1000
-10
-5
0
5
10
15
22���
collatz-bion-experiment-2018-10-31.nb

23. ������
��
First /@
����
GatherBy
#
2^exponent[#]
, exponent[#] & /@
���
Union[
���
Flatten[
�⋯
Table[
��
Reverse[2^
����
With[{li =
��
Mod[
�⋯
Log2[
���������������
NestWhileList[c2, n, # ≠ 1 &]], 1]},
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,

True}}] & /@
��
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{47 317, -14}, {120 149, -15}, {251, 1}, {167, 2}, {111, 3}, {175, 4}, {155, 5},
{103, 6}, {91, 7}, {107, 8}, {71, 9}, {47, 10}, {31, 11}, {27, 12}, {327, 13},
{1215, 14}, {1055, 15}, {703, 16}, {871, 17}, {3175, 18}, {3695, 19}, {2463, 20},
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31. 1
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collatz-bion-experiment-2018-10-31.nb

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9 37 149 597 2389 9557 38 229 152 917 611 669 2 446 677
11 45 181 725 2901 11 605 46 421 185 685 742 741 2 970 965
17 69 277 1109 4437 17 749 70 997 283 989 1 135 957 4 543 829
19 77 309 1237 4949 19 797 79 189 316 757 1 267 029 5 068 117
23 93 373 1493 5973 23 893 95 573 382 293 1 529 173 6 116 693
25 101 405 1621 6485 25 941 103 765 415 061 1 660 245 6 640 981
27 109 437 1749 6997 27 989 111 957 447 829 1 791 317 7 165 269
31 125 501 2005 8021 32 085 128 341 513 365 2 053 461 8 213 845
33 133 533 2133 8533 34 133 136 533 546 133 2 184 533 8 738 133
35 141 565 2261 9045 36 181 144 725 578 901 2 315 605 9 262 421
39 157 629 2517 10 069 40 277 161 109 644 437 2 577 749 10 310
41 165 661 2645 10 581 42 325 169 301 677 205 2 708 821 10 835
43 173 693 2773 11 093 44 373 177 493 709 973 2 839 893 11 359
47 189 757 3029 12 117 48 469 193 877 775 509 3 102 037 12 408
49 197 789 3157 12 629 50 517 202 069 808 277 3 233 109 12 932
51 205 821 3285 13 141 52 565 210 261 841 045 3 364 181 13 456
55 221 885 3541 14 165 56 661 226 645 906 581 3 626 325 14 505
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collatz-bion-experiment-2018-10-31.nb
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17 69 277 1109 4437 17 749
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23 93 373 1493 5973 23 893
25 101 405 1621 6485 25 941
27 109 437 1749 6997 27 989
31 125 501 2005 8021 32 085
33 133 533 2133 8533 34 133
35 141 565 2261 9045 36 181
39 157 629 2517 10 069 40 277
41 165 661 2645 10 581 42 325
43 173 693 2773 11 093 44 373
47 189 757 3029 12 117 48 469
49 197 789 3157 12 629 50 517
51 205 821 3285 13 141 52 565
55 221 885 3541 14 165 56 661
57 229 917 3669 14 677 58 709
59 237 949 3797 15 189 60 757
63 253 1013 4053 16 213 64 853
65 261 1045 4181 16 725 66 901
67 269 1077 4309 17 237 68 949
71 285 1141 4565 18 261 73 045
73 293 1173 4693 18 773 75 093
75 301 1205 4821 19 285 77 141
79 317 1269 5077 20 309 81 237
81 325 1301 5205 20 821 83 285
83 333 1333 5333 21 333 85 333
87 349 1397 5589 22 357 89 429
89 357 1429 5717 22 869 91 477
91 365 1461 5845 23 381 93 525
95 381 1525 6101 24 405 97 621
97 389 1557 6229 24 917 99 669
99 397 1589 6357 25 429 101 717
103 413 1653 6613 26 453 105 813
105 421 1685 6741 26 965 107 861
107 429 1717 6869 27 477 109 909
111 445 1781 7125 28 501 114 005
113 453 1813 7253 29 013 116 053
115 461 1845 7381 29 525 118 101
119 477 1909 7637 30 549 122 197
121 485 1941 7765 31 061 124 245
123 493 1973 7893 31 573 126 293
127 509 2037 8149 32 597 130 389
129 517 2069 8277 33 109 132 437
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collatz-bion-experiment-2018-10-31.nb

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115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157,
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7, 11, 15, 47, 1, 25, 17, 53, 9, 7, 19, 59, 5, 31, 21, 65, 11, 17, 23, 71, 3, 37,
25, 77, 13, 5, 27, 83, 7, 43, 29, 89, 15, 23, 31, 95, 1, 49, 33, 101, 17, 13, 35,
107, 9, 55, 37, 113, 19, 29, 39, 119, 5, 61, 41, 125, 21, 1, 43, 131, 11, 67, 45,
137, 23, 35, 47, 143, 3, 73, 49, 149, 25, 19, 51, 155, 13, 79, 53, 161, 27, 41,
55, 167, 7, 85, 57, 173, 29, 11, 59, 179, 15, 91, 61, 185, 31, 47, 63, 191, 1, 97,
65, 197, 33, 25, 67, 203, 17, 103, 69, 209, 35, 53, 71, 215, 9, 109, 73, 221, 37,
7, 75, 227, 19, 115, 77, 233, 39, 59, 79, 239, 5, 121, 81, 245, 41, 31, 83, 251,
21, 127, 85, 257, 43, 65, 87, 263, 11, 133, 89, 269, 45, 17, 91, 275, 23, 139,
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Plot[4.5 x, {x, 1, 80}],
�������
ListPlot[%]}]
������ {1, 11, 17, 5, 13, 29, 35, 19, 11, 47, 53, 7, 31, 65, 71, 37, 5, 83, 89,
23, 49, 101, 107, 55, 29, 119, 125, 1, 67, 137, 143, 73, 19, 155, 161, 41,
85, 173, 179, 91, 47, 191, 197, 25, 103, 209, 215, 109, 7, 227, 233, 59,
121, 245, 251, 127, 65, 263, 269, 17, 139, 281, 287, 145, 37, 299, 305}
������
20 40 60 80
50
100
150
200
250
300
350
������ {11, 17, 29, 35, 47, 53}
% //
��
Differences
������ {11, 17, 29, 35, 47, 53}
������ {6, 12, 6, 12, 6}
������
������
Table3 3 + (-1)n
, {n, 1, 10}
11 +
��
Accumulate[%]
������ {6, 12, 6, 12, 6, 12, 6, 12, 6, 12}
������ {17, 29, 35, 47, 53, 65, 71, 83, 89, 101}
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��
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Range[2, 203],
��������
MemberQ[{1, 5},
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{11, 17, 29, 35, 47, 53}
������ {2, 3, 6, 7, 10, 11}
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FindSequenceFunction[{2, 3, 6, 7, 10, 11}, n]
������
1
2
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38���
collatz-bion-experiment-2018-10-31.nb

39. ������
c2 /@ c2 /@
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��
Range[2, 203],
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MemberQ[{1, 5},
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������ {1, 17, 13, 1, 5, 11, 53, 29, 17, 71, 5, 11, 47, 49, 107, 7, 1, 125, 67,
35, 37, 19, 161, 83, 11, 179, 47, 1, 101, 103, 215, 55, 29, 233, 121,
31, 1, 65, 269, 137, 71, 287, 37, 19, 155, 157, 323, 41, 11, 341, 175,
89, 91, 23, 377, 191, 49, 395, 101, 13, 209, 211, 431, 109, 7, 449, 229}
������
10 20 30 40 50 60
100
200
300
400
������ {17, 53, 71, 107, 125, 161, 179}
% //
��
Differences
������ {17, 53, 71, 107, 125, 161, 179}
������ {36, 18, 36, 18, 36, 18}
������
������
Table-9 -3 + (-1)n
, {n, 1, 10}
17 +
��
Accumulate[%]
������ {36, 18, 36, 18, 36, 18, 36, 18, 36, 18}
������ {53, 71, 107, 125, 161, 179, 215, 233, 269, 287}
��
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FirstPosition[c2 /@ c2 /@
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Select[
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Range[2, 203],
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MemberQ[{1, 5},
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#] & /@ {17, 53, 71, 107, 125, 161, 179}
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FindSequenceFunction[
%,
n]
������ {2, 7, 10, 15, 18, 23, 26}
������
1
2
(-1)n
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collatz-bion-experiment-2018-10-31.nb
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40. ������
c2 /@ c2 /@ c2 /@
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Select[
��
Range[2, 203],
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MemberQ[{1, 5},
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Mod[#, 6]] &]
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ListPlot[%]
������ {1, 13, 5, 1, 1, 17, 5, 11, 13, 107, 1, 17, 71, 37, 161, 11, 1, 47,
101, 53, 7, 29, 121, 125, 17, 269, 71, 1, 19, 155, 323, 83, 11, 175,
91, 47, 1, 49, 101, 103, 107, 431, 7, 29, 233, 59, 485, 31, 17, 1, 263,
67, 137, 35, 283, 287, 37, 593, 19, 5, 157, 317, 647, 41, 11, 337, 43}
������
10 20 30 40 50 60
100
200
300
400
500
600
700
������ {13, 17, 107, 161, 269, 323, 431, 485}
% //
��
Differences
������ {13, 17, 107, 161, 269, 323, 431, 485}
������ {4, 90, 54, 108, 54, 108, 54}
������
������
Table27 3 + (-1)n
, {n, 1, 10}
53 +
��
Accumulate[%]
������ {54, 108, 54, 108, 54, 108, 54, 108, 54, 108}
������ {107, 215, 269, 377, 431, 539, 593, 701, 755, 863}
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First /@ 
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FirstPosition[
c2 /@ c2 /@ c2 /@
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Select[
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Range[2, 203],
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MemberQ[{1, 5},
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Mod[#, 6]] &],
#] & /@ {107, 161, 269, 323, 431, 485}
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FindSequenceFunction[
%,
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������ {10, 15, 26, 31, 42, 47}
������
1
2
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collatz-bion-experiment-2018-10-31.nb

41. ������
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Length[
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NestWhileList[c2, #, # ≠ 1 &]] & /@
������
Table
1
2
9 - 3 (-1)n
- 8 (-1)2 n
+ 16 n, {n, 20}
������ {3, 6, 4, 40, 3, 39, 7, 40, 8, 12, 6, 39, 4, 25, 7, 16, 5, 38, 8, 19}
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c2 /@ c2 /@ c2 /@ c2 /@
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Select[
��
Range[2, 500],
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MemberQ[{1, 5},
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������ {1, 5, 1, 1, 1, 13, 1, 17, 5, 161, 1, 13, 107, 7, 121, 17, 1, 71, 19, 5, 11, 11, 91, 47,
13, 101, 107, 1, 29, 233, 485, 125, 17, 263, 137, 71, 1, 37, 19, 155, 161, 647,
11, 11, 175, 89, 91, 47, 13, 1, 395, 101, 103, 53, 425, 431, 7, 445, 29, 1, 59,
119, 971, 31, 17, 253, 65, 17, 67, 269, 137, 35, 71, 1133, 1, 37, 593, 299, 607,
5, 5, 157, 319, 323, 41, 83, 167, 337, 11, 43, 175, 1, 179, 719, 1457, 23, 47,
749, 95, 1, 49, 49, 395, 199, 101, 1619, 103, 53, 209, 53, 425, 215, 1, 109, 881,
7, 449, 29, 911, 917, 233, 931, 59, 1, 479, 481, 1943, 245, 125, 31, 251, 17, 1,
1, 65, 65, 263, 2105, 67, 269, 1079, 271, 1093, 35, 71, 557, 281, 283, 143, 287,
577, 145, 37, 587, 593, 5, 19, 1205, 2429, 611, 155, 1235, 623, 157, 5, 317}
������
50 100 150
200
400
600
800
1000
1200
collatz-bion-experiment-2018-10-31.nb
���41

42. ������ {13, 17, 161, 233, 485, 647, 971, 1133, 1619, 1943, 2429}
��
Differences[%]
�������
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������ {13, 17, 161, 233, 485, 647, 971, 1133, 1619, 1943, 2429}
������ {4, 144, 72, 252, 162, 324, 162, 486, 324, 486}
������
2 4 6 8 10
500
1000
1500
2000
2500
������
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First /@ 
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FirstPosition[
c2 /@ c2 /@ c2 /@ c2 /@
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Select[
��
Range[2, 500],
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MemberQ[{1, 5},
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Mod[#, 6]] &],
#] & /@ {485, 647, 971, 1133, 1457, 1619}
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FindSequenceFunction[
%,
n]
������ {31, 42, 63, 74, 95, 106}
������
1
2
41 - 5 (-1)n
- 16 (-1)2 n
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������
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Table
1
2
1 - 2 (-1)2 n
+ (-1)1+n
+ 4 n, {n, 1, 50}
% //
��
Differences
������ {2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31,
34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66,
67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99}
������ {1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3,
1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1}
42���
collatz-bion-experiment-2018-10-31.nb

43. ������
������
Table
1
2
(-1)n
1 - 3 (-1)n
+ 8 (-1)n n, {n, 1, 50}
% //
��
Differences
������ {2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, 47, 50, 55, 58, 63, 66, 71, 74,
79, 82, 87, 90, 95, 98, 103, 106, 111, 114, 119, 122, 127, 130, 135, 138,
143, 146, 151, 154, 159, 162, 167, 170, 175, 178, 183, 186, 191, 194, 199}
������ {5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3,
5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5}
������
������
Table
1
2
9 - 3 (-1)n
- 8 (-1)2 n
+ 16 n, {n, 1, 50}
% //
��
Differences
������ {10, 15, 26, 31, 42, 47, 58, 63, 74, 79, 90, 95, 106, 111, 122, 127, 138, 143, 154,
159, 170, 175, 186, 191, 202, 207, 218, 223, 234, 239, 250, 255, 266, 271, 282,
287, 298, 303, 314, 319, 330, 335, 346, 351, 362, 367, 378, 383, 394, 399}
������ {5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5,
11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5, 11, 5}
������
������
Table
1
2
41 - 5 (-1)n
- 16 (-1)2 n
+ 32 n, {n, 1, 50}
% //
��
Differences
������ {31, 42, 63, 74, 95, 106, 127, 138, 159, 170, 191, 202, 223, 234, 255, 266, 287, 298,
319, 330, 351, 362, 383, 394, 415, 426, 447, 458, 479, 490, 511, 522, 543, 554,
575, 586, 607, 618, 639, 650, 671, 682, 703, 714, 735, 746, 767, 778, 799, 810}
������ {11, 21, 11, 21, 11, 21, 11, 21, 11, 21, 11, 21, 11, 21, 11,
21, 11, 21, 11, 21, 11, 21, 11, 21, 11, 21, 11, 21, 11, 21, 11, 21,
11, 21, 11, 21, 11, 21, 11, 21, 11, 21, 11, 21, 11, 21, 11, 21, 11}
������ {1, 3, 5, 11, 21}
������ {1, 3, 5, 11, 21}
collatz-bion-experiment-2018-10-31.nb
���43

44. ������ {2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31,
34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66,
67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99}
����
Intersection[{2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30,
31, 34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66,
67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99},
{2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, 47, 50, 55, 58, 63, 66, 71, 74, 79,
82, 87, 90, 95, 98, 103, 106, 111, 114, 119, 122, 127, 130, 135, 138, 143,
146, 151, 154, 159, 162, 167, 170, 175, 178, 183, 186, 191, 194, 199}]
����
Intersection[{2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30,
31, 34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66,
67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99},
{2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, 47, 50, 55, 58, 63, 66, 71, 74,
79, 82, 87, 90, 95, 98, 103, 106, 111, 114, 119, 122, 127, 130, 135, 138,
143, 146, 151, 154, 159, 162, 167, 170, 175, 178, 183, 186, 191, 194, 199},
{10, 15, 26, 31, 42, 47, 58, 63, 74, 79, 90, 95, 106, 111, 122, 127, 138, 143,
154, 159, 170, 175, 186, 191, 202, 207, 218, 223, 234, 239, 250, 255, 266, 271,
282, 287, 298, 303, 314, 319, 330, 335, 346, 351, 362, 367, 378, 383, 394, 399}]
������ {2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31,
34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66,
67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99}
������ {2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42,
47, 50, 55, 58, 63, 66, 71, 74, 79, 82, 87, 90, 95, 98}
������ {10, 15, 26, 31, 42, 47, 58, 63, 74, 79, 90, 95}
������
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Union[
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Range[2, 203],
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MemberQ[{1, 5},
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Mod[#, 6]] &]]]
% //
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ListPlot
������ {1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 47, 49, 53, 55, 59, 65, 67, 71, 73,
83, 85, 89, 91, 101, 103, 107, 109, 119, 121, 125, 127, 137, 139, 143, 145, 155,
161, 173, 179, 191, 197, 209, 215, 227, 233, 245, 251, 263, 269, 281, 287, 299, 305}
������
10 20 30 40 50
50
100
150
200
250
300
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collatz-bion-experiment-2018-10-31.nb

45. ������ (* ����������� *)
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Select[
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Range[203],
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MemberQ[{3},
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Mod[#, 6]] &]
������ {3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111,
117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201}
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Partition[
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Reverse[
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NestWhileList[c2, n, # ≠ 1 &]], 2, 1],
{n, 1, 500, 2},
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Length[#] ≥ 3 &
������ {{1 → 5, 1 → 21, 1 → 85, 1 → 341},
{5 → 3, 5 → 13, 5 → 53, 5 → 213}, {7 → 9, 7 → 37, 7 → 149},
{11 → 7, 11 → 29, 11 → 117, 11 → 469}, {13 → 17, 13 → 69, 13 → 277},
{17 → 11, 17 → 45, 17 → 181, 17 → 725}, {19 → 25, 19 → 101, 19 → 405},
{23 → 15, 23 → 61, 23 → 245}, {25 → 33, 25 → 133, 25 → 533},
{29 → 19, 29 → 77, 29 → 309}, {31 → 41, 31 → 165, 31 → 661},
{35 → 23, 35 → 93, 35 → 373}, {41 → 27, 41 → 109, 41 → 437},
{43 → 57, 43 → 229, 43 → 917}, {59 → 39, 59 → 157, 59 → 629},
{113 → 75, 113 → 301, 113 → 1205}, {125 → 83, 125 → 333, 125 → 1333},
{131 → 87, 131 → 349, 131 → 1397}, {167 → 111, 167 → 445, 167 → 1781},
{173 → 115, 173 → 461, 173 → 7381}, {205 → 273, 205 → 1093, 205 → 4373}}
������
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Length /@
�⋯
Table
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������ {8, 6, 20, 13, 27, 28, 122, 123, 36, 37, 157, 114, 53, 54, 99, 100, 101, 102, 103,
73, 167, 168, 169, 170, 171, 172, 248, 174, 188, 189, 252, 253, 179, 180, 318,
244, 196, 197, 154, 155, 156, 157, 401, 327, 496, 497, 162, 163, 332, 333}
������
10 20 30 40 50
100
200
300
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collatz-bion-experiment-2018-10-31.nb
���45

46. ������
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������ {3, 2, 7, 4, 9, 9, 45, 45, 11, 11, 57, 40, 16, 16, 33, 33, 33, 33,
33, 21, 57, 57, 57, 57, 57, 57, 86, 57, 62, 62, 86, 86, 57, 57, 110,
81, 62, 62, 45, 45, 45, 45, 139, 110, 175, 175, 45, 45, 110, 110}
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179, 304, 305, 474, 475, 445, 446, 385, 386, 449, 450, 451, 452, 528, 529, 530,
531, 532, 533, 534, 535, 536, 537, 587, 588, 589, 590, 591, 592, 593, 594}
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10 20 30 40 50
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110, 110, 175, 175, 163, 163, 139, 139, 163, 163, 163, 163, 192, 192, 192,
192, 192, 192, 192, 192, 192, 192, 211, 211, 211, 211, 211, 211, 211, 211}
������
10 20 30 40 50
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collatz-bion-experiment-2018-10-31.nb

47. ������
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NestWhileListc2, 27
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���������������
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- 1, # ≠ 1 &
������ {127, 191, 287, 431, 647, 971, 1457, 1093, 205, 77, 29, 11, 17, 13, 5, 1}
������ {255, 383, 575, 863, 1295, 1943, 2915, 4373, 205, 77, 29, 11, 17, 13, 5, 1}
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255
4
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collatz-bion-experiment-2018-10-31.nb
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48. ������
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15
7
46
22
23
11
70
34
35
17
106
52
53
26
160
13
80
40
40
20
20
10
63
31
190
94
95
47
286
142
143
71
430
214
215
107
646
322
323
161
970
484
485
242
255
127
766
382
383
191
1150
574
575
287
1726
862
863
431
2590
1294
1295
647
3886
1942
1943
971
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191
575
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863
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2915
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4373
1093
205
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312, 257, 255, 448, 446, 444, 406, 440, 438, 436, 434, 432, 430, 428, 414, 412,
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53 → {53, 5}, 59 → {59, 89, 67, 101, 19, 29, 11, 17, 13, 5},
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