collatz-bion-experiment-2018-10-31

collatz-bion-experiment-2018-10-31.nb
created at 2018-10-31
(* �� *)
c[n_] :=
n
2
Mod[n, 2] ⩵ 0
3 n + 1 True
���� (* �� *)
c2[n_] :=
n
2^IntegerExponent[n,2]
Mod[n, 2] ⩵ 0
3 n+1
2^IntegerExponent[3 n+1,2]
True
����
�⋯
Plot[
��
Mod[x, 1], {x, 0, 1}]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
��
Log2[c[n]], 1]}, {n, 1, 256}]]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0

View Slide

����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
��
Log2[c2[n]], 1]}, {n, 1, 256}]]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
��
Log2[3 n + 1], 1]}, {n, 1, 256, 2}]]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
����
�⋯
Plot[
��
Mod[x +
��
Log2[3], 1], {x, 0, 1}]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
2��

View Slide

����
�⋯
Plot[
��
Mod[x + 2
��
Log2[3], 1], {x, 0, 1}]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
��
Log2[c[c[n]]], 1]}, {n, 1, 256}]]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
��
Log2[c2[c2[n]]], 1]}, {n, 1, 256}]]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
��3

View Slide

����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
�⋯
Log2[
��
Nest[c, n, 10]], 1]}, {n, 1, 1024}]]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
�⋯
Log2[
��
Nest[c, n, 100]], 1]}, {n, 1, 1024}]]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
�⋯
Log2[
��
Nest[c, n, 1000]], 1]}, {n, 1, 1024}]]
����
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
4��

View Slide

����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
�⋯
Log2[
��
Nest[c2, n, 10]], 1]}, {n, 1, 1024}]]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
�⋯
Log2[
��
Nest[c2, n, 40]], 1]}, {n, 1, 1024}]]
����
0.2 0.4 0.6 0.8 1.0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
�⋯
Log2[
��
Nest[c2, n, 60]], 1]}, {n, 1, 1024}]]
����
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
��5

View Slide

����
��⋯
ListPlot[
��⋯
Table[{
��
Mod[
��
Log2[n], 1],
��
Mod[
�⋯
Log2[
��
Nest[c2, n, 80]], 1]}, {n, 1, 1024}]]
����
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
����
��
Mod[
�⋯
Log2[
��
NestWhileList[c, 7, # ≠ 1 &]], 1] //
��
FullSimplify
��
DeleteDuplicates[%]
���� 
Log
7
4

Log[2]
,
Log
11
8

Log[2]
,
Log
11
8

Log[2]
,
Log
17
16

Log[2]
,
Log
17
16

Log[2]
,
Log
13
8

Log[2]
,
Log
13
8

Log[2]
,
Log
13
8

Log[2]
,
Log
5
4

Log[2]
,
Log
5
4

Log[2]
,
Log
5
4

Log[2]
,
Log
5
4

Log[2]
, 0, 0, 0, 0, 0
���� 
Log
7
4

Log[2]
,
Log
11
8

Log[2]
,
Log
17
16

Log[2]
,
Log
13
8

Log[2]
,
Log
5
4

Log[2]
, 0
����
��
NestWhileList[c, 27, # ≠ 1 &] //
��
ListPlot
����
20 40 60 80 100
500
1000
1500
2000
2500
3000
3500
6��

View Slide

����
��
Mod[
�⋯
Log2[
��
NestWhileList[c, 27, # ≠ 1 &]], 1] //
��
ListPlot
����
20 40 60 80 100
0.2
0.4
0.6
0.8
1.0
���� (* 0�1�� *)
��
Mod[
�⋯
Log2[
��
NestWhileList[c2, 7, # ≠ 1 &]], 1] //
��
FullSimplify
% //
��
ListPlot
���� 
Log
7
4

Log[2]
,
Log
11
8

Log[2]
,
Log
17
16

Log[2]
,
Log
13
8

Log[2]
,
Log
5
4

Log[2]
, 0
����
1 2 3 4 5 6
0.2
0.4
0.6
0.8
��7

View Slide

���� (* �� *)
2^ 1 +
Log
7
4

Log[2]
, 1 +
Log
11
8

Log[2]
, 1 +
Log
17
16

Log[2]
,
Log
13
8

Log[2]
,
Log
5
4

Log[2]
, 0 //
��
FullSimplify
% //
��
ListPlot
���� 
7
2
,
11
4
,
17
8
,
13
8
,
5
4
, 1
����
1 2 3 4 5 6
0.5
1.0
1.5
2.0
2.5
3.0
3.5
8��

View Slide

���� (* �� *)
��
Mod[
�⋯
Log2[
��
NestWhileList[c2, 27, # ≠ 1 &]], 1] //
��
FullSimplify
% //
��
ListPlot
���� 
Log
27
16

Log[2]
,
Log
41
32

Log[2]
,
Log
31
16

Log[2]
,
Log
47
32

Log[2]
,
Log
71
64

Log[2]
,
Log
107
64

Log[2]
, -7 +
Log[161]
Log[2]
,
Log
121
64

Log[2]
,
Log
91
64

Log[2]
, -7 +
Log[137]
Log[2]
,
Log
103
64

Log[2]
, -7 +
Log[155]
Log[2]
, -7 +
Log[233]
Log[2]
,
-7 +
Log[175]
Log[2]
, -8 +
Log[263]
Log[2]
, -8 +
Log[395]
Log[2]
, -9 +
Log[593]
Log[2]
, -8 +
Log[445]
Log[2]
,
-7 +
Log[167]
Log[2]
, -7 +
Log[251]
Log[2]
, -8 +
Log[377]
Log[2]
, -8 +
Log[283]
Log[2]
, -8 +
Log[425]
Log[2]
,
-8 +
Log[319]
Log[2]
, -8 +
Log[479]
Log[2]
, -9 +
Log[719]
Log[2]
, -10 +
Log[1079]
Log[2]
,
-10 +
Log[1619]
Log[2]
, -11 +
Log[2429]
Log[2]
, -9 +
Log[911]
Log[2]
, -10 +
Log[1367]
Log[2]
,
-11 +
Log[2051]
Log[2]
, -11 +
Log[3077]
Log[2]
, -9 +
Log[577]
Log[2]
, -8 +
Log[433]
Log[2]
,
-8 +
Log[325]
Log[2]
,
Log
61
32

Log[2]
,
Log
23
16

Log[2]
,
Log
35
32

Log[2]
,
Log
53
32

Log[2]
,
Log
5
4

Log[2]
, 0
����
10 20 30 40
0.2
0.4
0.6
0.8
1.0
���� (* �� *)
��
Differences[
⋯
N[
��
Mod[
�⋯
Log2[
��
NestWhileList[c2, 27, # ≠ 1 &]], 1]]]
��
Select[%, # > 0 &] //
��
Mean
���� {-0.397335, 0.596644, -0.399607, -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}
����
0.588313
��9

View Slide

����
��
Log2
3
2
 //
��
N
����
0.584963
����
��
Accumulate[
��
Mod[
��
Differences[
��
Mod[
�⋯
Log2[
��
NestWhileList[c2, 27, # ≠ 1 &]], 1] //
��
N], 1]];
��
ListPlot[%]
��
Table%%〚n〛 -
��
Log2
3
2
 n, {n, 1, 41} //
��
ListPlot
����
10 20 30 40
5
10
15
20
25
����
10 20 30 40
0.05
0.10
0.15
10��

View Slide

����
��
Reverse
2^
��
Accumulate[
��
Mod[
��
Differences[
��
Reverse[
��
Mod[
�⋯
Log2[
��
NestWhileList[c2, 27, # ≠ 1 &]],
1]]], 1]] //
��
FullSimplify 
�
Ratios[
%] //
��
ListPlot
���� 110 592, 83 968, 63 488, 48 128, 36 352, 27 392, 20 608, 15 488, 11 648, 8768, 6592,
4960, 3728, 2800, 2104, 1580, 1186, 890, 668, 502, 377, 283,
425
2
,
319
2
,
479
4
,
719
8
,
1079
16
,
1619
32
,
2429
64
,
911
32
,
1367
64
,
2051
128
,
3077
256
,
577
64
,
433
64
,
325
64
,
61
16
,
23
8
,
35
16
,
53
32
,
5
4

����
10 20 30 40
0.750
0.752
0.754
0.756
0.758
0.760
���� (* �� *)
��
Tablen → 
��
NestWhileList[c2, n, # ≠ 1 &] //
��
Length, {n, 1, 10 000, 2};
��⋯
SortBy[%,
��
Last[#] &] //
��
Last
����
6171 → 97
����
��
Mod[
�⋯
Log2[
��
NestWhileList[c2, 6171, # ≠ 1 &]], 1] //
��
ListPlot
����
20 40 60 80 100
0.2
0.4
0.6
0.8
1.0
��11

View Slide

��
Mod[
�⋯
Log2[
��
NestWhileList[c2, 6171, # ≠ 1 &]], 1] //
��
ListPlot
����
20 40 60 80 100
0.2
0.4
0.6
0.8
1.0
����
��
Differences[
⋯
N[
��
Mod[
�⋯
Log2[
��
NestWhileList[c2, 6171, # ≠ 1 &]], 1]]]
��
Select[%, # > 0 &] //
��
Mean
���� {-0.41496, 0.585014, -0.414968, 0.585009, -0.415007, -0.415017, 0.584976,
-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}
����
0.586408
12��

View Slide

����
��
Accumulate[
��
Mod[
��
Differences[
��
Mod[
�⋯
Log2[
��
NestWhileList[c2, 6171, # ≠ 1 &]], 1] //
��
N], 1]];
��
ListPlot[%]
��
Table%%〚n〛 -
��
Log2
3
2
 n, {n, 1, 96} //
��
ListPlot
����
20 40 60 80
10
20
30
40
50
60
����
20 40 60 80
0.05
0.10
0.15
0.20
0.25
��13

View Slide

���� (* �� *)
��
With[{li =
��
Mod[
�⋯
Log2[
��
NestWhileList[c2, 6171, # ≠ 1 &]], 1]},
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,
�
True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]] //
��
ListPlot
����
20 40 60 80 100
10
20
30
40
����
1 
��
NestWhileList[c2, 6171, # ≠ 1 &] //
��
ListLogPlot
����
20 40 60 80 100
10-5
10-4
0.001
0.01
0.1
1
14��

View Slide

����
��
Reverse2^
��
Accumulate[
��
Mod[
��
Differences[
��
Reverse[
��
Mod[
�⋯
Log2[
��
NestWhileList[c2, 6171, # ≠ 1 &]], 1]]], 1]] //
��
FullSimplify 
�
Ratios[
%] //
��
ListPlot
���� 828 257 599 488, 621 226 754 048, 465 936 842 752, 349 469 409 280, 262 110 445 568,
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,
64 000, 48 128, 36 352, 27 392, 20 608, 15 488, 11 648, 8768, 6592, 4960, 3728,
2800, 2104, 1580, 1186, 890, 668, 502, 377, 283,
425
2
,
319
2
,
479
4
,
719
8
,
1079
16
,
1619
32
,
2429
64
,
911
32
,
1367
64
,
2051
128
,
3077
256
,
577
64
,
433
64
,
325
64
,
61
16
,
23
8
,
35
16
,
53
32
,
5
4

����
20 40 60 80
0.750
0.751
0.752
0.753
0.754
��15

View Slide

���� (* c2�� *)
�⋯
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}] //
��
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��

View Slide

(* �� *)
��
Union[
��
Flatten[
�⋯
Table[
��
Reverse[2^
��
With[{li =
��
Mod[
�⋯
Log2[
��
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}
��17

View Slide

���� {3, 13, 53, 113, 227}
��
Hyperlink["https://oeis.org/A198584"]
���� {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[
��
��
First /@
��
Select[%,
��
Last[#] ⩵ 1 &]
���� {1}
����
��
Table[n →
��
Length[
��
��
First /@
��
Select[%,
��
Last[#] ⩵ 2 &]
���� {5, 21, 85, 341}
����
��
Table[n →
��
Length[
��
��
First /@
��
Select[%,
��
Last[#] ⩵ 10 &]
���� {43, 87, 89, 173, 177, 179}
18��

View Slide

(* �� *)
��⋯
SortBy[
��
Union[
��
Flatten[
�⋯
Table[
��
With[{processLi =
��
NestWhileList[c2, n, # ≠ 1 &]},
��
With[{valueLi = 2^
��
With[{li =
��
Mod[
��
Log2[processLi], 1]},
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,
�
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 =
��
��
With[{valueLi =
��
Reverse[2^
��
With[{li =
��
Mod[
��
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 /@
%%
��19

View Slide

���� 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��

View Slide

���� 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[
��
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}}
��21

View Slide

���� (* �� *)
��
SortBy
#
2^exponent[#]
��
Union[
��
Flatten[
�⋯
Table[
��
Reverse[2^
��
With[{li =
��
Mod[
�⋯
Log2[
��
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��

View Slide

����
��
First /@
��
GatherBy
#
2^exponent[#]
��
Union[
��
Flatten[
�⋯
Table[
��
Reverse[2^
��
With[{li =
��
Mod[
�⋯
Log2[
��
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,
�
True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]], {n, 1, 10 000, 2}]]],
��
Last
���� {{1, 0}, {5, -2}, {21, -4}, {85, -6}, {341, -8}, {1365, -10}, {5461, -12},
{3, -1}, {13, -3}, {53, -5}, {213, -7}, {853, -9}, {3413, -11}, {14 549, -13},
{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},
{2919, 21}, {8351, 22}, {5567, 23}, {3711, 24}, {7963, 25}, {6943, 26}, {6171, 27}}
���� (* n��1�� *)
{251, 167, 111, 175, 155, 103, 91, 107, 71, 47, 31, 27, 327, 1215, 1055} //
��
ListPlot
����
2 4 6 8 10 12 14
200
400
600
800
��23

View Slide

����
�⋯
Table
��
SortBy
#
2^exponent[#]
��
Union[
��
Flatten[
�⋯
Table[
��
Reverse[2^
��
With[{li =
��
Mod[
�⋯
Log2[
��
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,
�
True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]],
{n, 1, k, 2}]]],
��
First, {k, {100, 1000, 10 000}};
��
First /@
��
Select[#,
��
Last[#] > 0 &] & /@ %;
��
Show[
��
Join[{
��
Plot[4.5 x, {x, 1, 3000}],
��
Plot[5 x, {x, 1, 3000}]},
��
ListPlot /@ %]]
����
500 1000 1500 2000 2500 3000
2000
4000
6000
8000
10000
12000
14000
����
��
Floor2
��
Log23 × 3077 × 2-8
- 2
��
��
Length
����
10
����
10
����
��
Floor2
��
Log23 × 449 × 28
- 2
��
��
Length
����
36
����
42
24��

View Slide

����
��
Table
��
Numerator /@ 
��
Reverse[2^
��
With[{li =
��
Mod[
�⋯
Log2[
��
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,
�
True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]], {n, 1, 20, 2} //
��
TableForm
����
1
1 5 3
1 5
1 5 13 17 11 7
1 5 13 17 11 7 9
1 5 13 17 11
1 5 13
1 5 53 35 23 15
1 5 13 17
1 5 13 17 11 29 19
����
�⋯
Table[
��
Reverse[
��
NestWhileList
[c2, n, # ≠ 1 &]], {n, 1, 20, 2}] //
��
TableForm
����
1
1 5 3
1 5
1 5 13 17 11 7
1 5 13 17 11 7 9
1 5 13 17 11
1 5 13
1 5 53 35 23 15
1 5 13 17
1 5 13 17 11 29 19
����
��
Union
��
Flatten
��⋯
Table
��
Rule @@ # & /@
��⋯
Partition[
��
Reverse[2^
��
With[{li =
��
Mod[
�⋯
Log2[
��
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,
�
True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]], 2, 1], {n, 1, 20, 2}
���� 1 →
5
4
,
5
4
→
3
2
,
5
4
→
13
8
,
5
4
→
53
32
,
13
8
→
17
8
,
53
32
→
35
16
,
17
8
→
11
4
,
35
16
→
23
8
,
11
4
→
7
2
,
11
4
→
29
8
,
23
8
→
15
4
,
7
2
→
9
2
,
29
8
→
19
4

��25

View Slide

����
��⋯
TreePlot
��
Union
��
Flatten
��⋯
Table
��
Rule @@ # & /@
��⋯
Partition[
��
Reverse[2^
��
With[{li =
��
Mod[
�⋯
Log2[
��
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,
�
True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]], 2, 1],
{n, 1, 100, 2},
��
Bottom, 1,
��
VertexLabels →
��
Automatic
3728
4960
6592
8768
11648
15488
15520
20608
20672
27392
27520
36352
36608
48128
48640
63488
64000
64512
83968
84992
110592
112640
149504
198656
26��

View Slide

����
39
2
79
4
1367
64
911
32
2429
64
1619
32
1079
16
719
8
479
4
319
2
425
2
283
377
502
668
890
1186
1580
2104
2800
��27

View Slide

1
5
4
21
16
85
64
3
2
13
8
53
32
113
64
17
8
69
32
35
16
75
32
11
4
45
16
23
8
93
32
7
2
29
8
15
4
61
16
9
2
37
8
149
32
19
4
77
16
81
16
325
64
49
8
197
32
99
16
25
4
101
16
51
8
433
64
65
8
131
16
33
4
67
8
269
32
577
64
43
4
87
8
89
8
179
16
3077
256
57
4
59
4
119
8
2051
128
����
��⋯
TreePlot
��
Union
��
Flatten
��⋯
Table
��
Rule @@ # & /@
��⋯
Partition[
��
Reverse[
��
NestWhileList[c2, n, # ≠ 1 &]], 2, 1],
{n, 1, 100, 2},
��
Bottom, 1,
��
VertexLabels →
��
Automatic
31
41
27
125
55
73
97
83
63
28��

View Slide

����
47
71
91
121
485
95
103
137
107
161
143
155
167
445
175
233
215
251
263
283
377
323
395
593
��29

View Slide

5
21
85
3
13
53
7
9
37
149
11
29
17
69
45
19
25
101
23
15
61
33
77
35
93
49
197
43
57
65
59
39
81
325
67
89
51
113
269
75
119
79
131
87
99
179
319
425
433
577
479
3077
719
911
2429
1079
1367
1619
2051
30��

View Slide

1
5
21
85
����
��
Table
4n
- 1
3
, {n, 1, 10}
c2 /@ %
���� {1, 5, 21, 85, 341, 1365, 5461, 21 845, 87 381, 349 525}
���� {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}
����
��
Table
10 × 4n
- 1
3
, {n, 0, 10}
c2 /@ %
���� {3, 13, 53, 213, 853, 3413, 13 653, 54 613, 218 453, 873 813, 3 495 253}
���� {5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5}
����
��
Table
7 × 4n
- 1
3
, {n, 1, 10}
c2 /@ %
���� {9, 37, 149, 597, 2389, 9557, 38 229, 152 917, 611 669, 2 446 677}
���� {7, 7, 7, 7, 7, 7, 7, 7, 7, 7}
����
��
Table
11 × 4n
- 2
6
, {n, 1, 10}
c2 /@ %
���� {7, 29, 117, 469, 1877, 7509, 30 037, 120 149, 480 597, 1 922 389}
���� {11, 11, 11, 11, 11, 11, 11, 11, 11, 11}
����
��
Table
13 × 4n
- 1
3
, {n, 1, 10}
c2 /@ %
���� {17, 69, 277, 1109, 4437, 17 749, 70 997, 283 989, 1 135 957, 4 543 829}
���� {13, 13, 13, 13, 13, 13, 13, 13, 13, 13}
����
��
Table
17 × 4n
- 2
6
, {n, 1, 10}
c2 /@ %
���� {11, 45, 181, 725, 2901, 11 605, 46 421, 185 685, 742 741, 2 970 965}
���� {17, 17, 17, 17, 17, 17, 17, 17, 17, 17}
��31

View Slide

���� {11, 45, 181, 181 × 4 + 1}
��
FindSequenceFunction[%, n]
���� {11, 45, 181, 725}
����
1
6
-2 + 17 × 22 n

���� {
��
NestList[4 # + 1 &, 1, 10],
��
NestList[4 # + 1 &, 3, 10],
��
NestList[4 # + 1 &, 7, 10],
��
NestList[4 # + 1 &, 9, 10],
��
NestList[4 # + 1 &, 11, 10],
��
NestList[4 # + 1 &, 17, 10],
��
NestList[4 # + 1 &, 19, 10],
��
NestList[4 # + 1 &, 23, 10],
��
NestList[4 # + 1 &, 25, 10],
��
NestList[4 # + 1 &, 27, 10],
��
NestList[4 # + 1 &, 31, 10],
��
NestList[4 # + 1 &, 33, 10],
��
NestList[4 # + 1 &, 35, 10],
��
NestList[4 # + 1 &, 39, 10],
��
NestList[4 # + 1 &, 41, 10],
��
NestList[4 # + 1 &, 43, 10],
��
NestList[4 # + 1 &, 47, 10],
��
NestList[4 # + 1 &, 49, 10],
��
NestList[4 # + 1 &, 51, 10],
��
NestList[4 # + 1 &, 55, 10]
} //
��
TableForm
32��

View Slide

����
1 5 21 85 341 1365 5461 21 845 87 381 349 525
3 13 53 213 853 3413 13 653 54 613 218 453 873 813
7 29 117 469 1877 7509 30 037 120 149 480 597 1 922 389
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
����
li4np1[init_, n_] :=
��
NestList[4 # + 1 &, init, n]
��33

View Slide

����
tbl = {li4np1[1, 5]};
⋯
Do[
��
AppendTo[tbl, li4np1[
��
Min[
��
Complement[
��
Range[1,
��
Max[
��
Flatten[tbl]], 2],
��
Flatten[tbl]]], 5]], {50}]
tbl //
��
TableForm
����
1 5 21 85 341 1365
3 13 53 213 853 3413
7 29 117 469 1877 7509
9 37 149 597 2389 9557
11 45 181 725 2901 11 605
15 61 245 981 3925 15 701
17 69 277 1109 4437 17 749
19 77 309 1237 4949 19 797
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
131 525 2101 8405 33 621 134 485
135 541 2165 8661 34 645 138 581
34��

View Slide

����
��
First[tbl]
% //
��
ListPlot
%% //
��
Differences
���� {1, 3, 7, 9, 11, 15, 17, 19, 23, 25, 27, 31, 33, 35, 39, 41, 43, 47, 49,
51, 55, 57, 59, 63, 65, 67, 71, 73, 75, 79, 81, 83, 87, 89, 91, 95, 97,
99, 103, 105, 107, 111, 113, 115, 119, 121, 123, 127, 129, 131, 135}
����
10 20 30 40 50
20
40
60
80
100
120
140
���� {2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2,
4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4}
����
��
FindSequenceFunction[{2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2,
4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4}, n]
����
Mod2 + 2 n + n2, 3, 2
����
c2 /@
��
Prepend1 +
��
Accumulate
�⋯
Table
��
Modn2
+ 2 n + 2, 3, 2, {n, 1, 20}, 1
��
Sort[%]
���� {1, 5, 11, 7, 17, 23, 13, 29, 35, 19, 41, 47, 25, 53, 59, 31, 65, 71, 37, 77, 83}
���� {1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 47, 53, 59, 65, 71, 77, 83}
����
c2 /@
��
Prepend1 +
��
Accumulate
�⋯
Table
��
Modn2
+ 2 n + 2, 3, 2, {n, 1, 40}, 1
��
Sort[%]
���� {1, 5, 11, 7, 17, 23, 13, 29, 35, 19, 41, 47, 25, 53, 59, 31, 65, 71, 37, 77, 83, 43, 89,
95, 49, 101, 107, 55, 113, 119, 61, 125, 131, 67, 137, 143, 73, 149, 155, 79, 161}
���� {1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67,
71, 73, 77, 79, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161}
��35

View Slide

����
c2 /@
��
Prepend1 +
��
Accumulate
�⋯
Table
��
Modn2
+ 2 n + 2, 3, 2, {n, 1, 1000}, 1;
��
Sort[%];
% //
��
ListPlot
����
200 400 600 800 1000
1000
2000
3000
4000
����
��
��
Select[
��
Range[203],
��
MemberQ[{1, 5},
��
Mod[#, 6]] &]
����
���� {1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59,
61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113,
115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157,
161, 163, 167, 169, 173, 175, 179, 181, 185, 187, 191, 193, 197, 199, 203}
36��

View Slide

����
c2 /@
��
Range[2, 203]
��
ListPlot[%]
��
���� {1, 5, 1, 1, 3, 11, 1, 7, 5, 17, 3, 5, 7, 23, 1, 13, 9, 29, 5, 1, 11, 35, 3, 19, 13, 41,
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,
93, 281, 47, 71, 95, 287, 3, 145, 97, 293, 49, 37, 99, 299, 25, 151, 101, 305}
����
50 100 150 200
50
100
150
200
250
300
����
���� {5, 11, 17, 23, 29}
���� {5, 11, 17, 23, 29}
����
��
Table[6 n - 1, {n, 1, 10}]
% //
��
Differences
���� {5, 11, 17, 23, 29, 35, 41, 47, 53, 59}
���� {6, 6, 6, 6, 6, 6, 6, 6, 6}
��37

View Slide

����
c2 /@
��
Select[
��
Range[2, 203],
��
MemberQ[{1, 5},
��
Mod[#, 6]] &]
��
Show[{
��
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}
����
��
First /@

��
FirstPosition[c2 /@
��
Select[
��
Range[2, 203],
��
MemberQ[{1, 5},
��
Mod[#, 6]] &], #] & /@
{11, 17, 29, 35, 47, 53}
���� {2, 3, 6, 7, 10, 11}
����
��
FindSequenceFunction[{2, 3, 6, 7, 10, 11}, n]
����
1
2
1 - 2 (-1)2 n
+ (-1)1+n
+ 4 n
38��

View Slide

����
c2 /@ c2 /@
��
Select[
��
Range[2, 203],
��
MemberQ[{1, 5},
��
Mod[#, 6]] &]
��
ListPlot[%]
���� {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}
��
First /@

��
FirstPosition[c2 /@ c2 /@
��
Select[
��
Range[2, 203],
��
MemberQ[{1, 5},
��
Mod[#, 6]] &],
#] & /@ {17, 53, 71, 107, 125, 161, 179}
��
FindSequenceFunction[
%,
n]
���� {2, 7, 10, 15, 18, 23, 26}
����
1
2
(-1)n
1 - 3 (-1)n
+ 8 (-1)n n
��39

View Slide

����
c2 /@ c2 /@ c2 /@
��
Select[
��
Range[2, 203],
��
MemberQ[{1, 5},
��
Mod[#, 6]] &]
��
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}
����
��
First /@ 
��
FirstPosition[
c2 /@ c2 /@ c2 /@
��
Select[
��
Range[2, 203],
��
MemberQ[{1, 5},
��
Mod[#, 6]] &],
#] & /@ {107, 161, 269, 323, 431, 485}
��
%,
n]
���� {10, 15, 26, 31, 42, 47}
����
1
2
9 - 3 (-1)n
- 8 (-1)2 n
+ 16 n
40��

View Slide

����
��
Length[
��
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}
����
c2 /@ c2 /@ c2 /@ c2 /@
��
Select[
��
Range[2, 500],
��
MemberQ[{1, 5},
��
Mod[#, 6]] &]
��
ListPlot[%]
���� {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
��41

View Slide

���� {13, 17, 161, 233, 485, 647, 971, 1133, 1619, 1943, 2429}
��
Differences[%]
��
ListPlot[%%]
���� {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
����
��
First /@ 
��
FirstPosition[
c2 /@ c2 /@ c2 /@ c2 /@
��
Select[
��
Range[2, 500],
��
MemberQ[{1, 5},
��
Mod[#, 6]] &],
#] & /@ {485, 647, 971, 1133, 1457, 1619}
��
%,
n]
���� {31, 42, 63, 74, 95, 106}
����
1
2
41 - 5 (-1)n
- 16 (-1)2 n
+ 32 n
����
��
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��

View Slide

����
��
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}
��43

View Slide

���� {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}
����
��
Union[
��
Sort[c2 /@
��
Select[
��
Range[2, 203],
��
MemberQ[{1, 5},
��
Mod[#, 6]] &]]]
% //
��
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
44��

View Slide

���� (* �� *)
��
Select[
��
Range[203],
��
MemberQ[{3},
��
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}
����
��
Select
��
GatherBy
��
Union
��
Flatten
��⋯
Table
��
Rule @@ # & /@
��⋯
Partition[
��
Reverse[
��
NestWhileList[c2, n, # ≠ 1 &]], 2, 1],
{n, 1, 500, 2},
��
First,
��
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}}
����
��
Length /@
�⋯
Table
��
NestWhileListc, 2n
+ 1, # ≠ 1 &, {n, 1, 50}
% //
��
ListPlot
��
���� {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
400
500
����
��45

View Slide

����
��
Length /@
�⋯
Table
��
NestWhileListc2, 2n
+ 1, # ≠ 1 &, {n, 1, 50}
���� {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}
����
��
Length /@
�⋯
Table
��
NestWhileListc, 2n
- 1, # ≠ 1 &, {n, 1, 50}
% //
��
ListPlot
��
���� {1, 8, 17, 18, 107, 108, 47, 48, 62, 63, 157, 158, 159, 160, 130, 131, 225, 226, 178,
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}
����
10 20 30 40 50
100
200
300
400
500
600
����
����
��
Length /@
�⋯
Table
��
- 1, # ≠ 1 &, {n, 1, 50}
% //
��
ListPlot
���� {1, 3, 6, 6, 40, 40, 16, 16, 21, 21, 57, 57, 57, 57, 45, 45, 81, 81, 62, 62,
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
50
100
150
200
46��

View Slide

����
��
NestWhileListc2, 27
- 1, # ≠ 1 &
��
NestWhileListc2, 28
- 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}
����
��
Select[
��
Union[
��
Flatten[
�⋯
Table[
��
With[{processLi =
��
��
With[{valueLi = 2^
��
With[{li =
��
Mod[
��
li -
��
Append[
��
Reverse[
��
Accumulate[
��
Piecewise[{{0, # ≥ 0}, {-1,
�
True}}] & /@
��
Differences[
��
Reverse[li]]]], 0]]},
��
Thread[processLi → valueLi]]],
{n, 1, 5000, 2}]]],
��⋯
IntegerQ[
�⋯
Log2[
��
First[#] + 1]] &]
���� 1 → 1, 3 →
3
2
, 7 →
7
2
, 15 →
15
4
, 31 → 63 488, 63 → 64 512, 127 →
127
2
, 255 →
255
4
,
511 →
511
2
, 1023 →
1023
4
, 2047 → 8 384 512, 4095 → 8 386 560, 8191 → 8 387 584
����
15
4

7
2
64 512  63 488
255
4

127
2
1023
4

511
2
8 386 560  8 384 512
����
15
14
����
63
62
����
255
254
����
1023
1022
����
4095
4094
��47

View Slide

����
��
Table

��
NestWhileListc, 22 n
- 1, # ≠ 1 & //
�⋯
Most,
��
NestWhileListc, 22 n-1
- 1, # ≠ 1 &,
{n, 2, 5} //
��
TableForm
����
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
1023
511
3070
1534
1535
767
4606
2302
2303
1151
6910
3454
3455
1727
10 366
5182
5183
2591
15 550
7774
7775
3887
����
��⋯
Table
��
NestWhileListc2, 22 n
- 1, # ≠ 1 &,
��
NestWhileListc2, 22 n-1
- 1, # ≠ 1 &,
{n, 2, 5} //
��
TableForm
����
15
7
23
11
35
17
53
13
5
5
1
1
63
31
95
47
143
71
215
107
323
161
485
121
91
91
137
137
103
103
155
155
255
127
383
191
575
287
863
431
1295
647
1943
971
2915
1457
4373
1093
205
205
77
77
1023
511
1535
767
2303
1151
3455
1727
5183
2591
7775
3887
11 663
5831
17 495
8747
26 243
13 121
39 365
9841
���� {{17, 1}, {19, 1}}
����
�⋯
Table
⋯
N
��
Divide @@@
��
Select

��
- 1, # ≠ 1 &,
��
- 1, # ≠ 1 &,
#〚1〛 ≠ #〚2〛 &, 3, {n, 2, 10} //
��
TableForm
����
2.14 2.09 2.06 4.08
2.03 2.02 2.01 2.01 2.01 4.01
2.01 2.01 2.00 2.00 2.00 2.00 2.00 4.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 4.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
48��

View Slide

����
�⋯
Table
��
Last
��
Select

��
- 1, # ≠ 1 &,
��
- 1, # ≠ 1 &,
#〚1〛 ≠ #〚2〛 &, {n, 2, 10} //
��
TableForm
����
53 13
485 121
4373 1093
39 365 9841
354 293 88 573
3 188 645 797 161
28 697 813 7 174 453
258 280 325 64 570 081
2 324 522 933 581 130 733
���� {13, 121, 1093, 9841, 88 573, 797 161, 7 174 453, 64 570 081, 581 130 733}
��
Table
3 × 9n
- 1
2
, {n, 1, 9}
��
���� {13, 121, 1093, 9841, 88 573, 797 161, 7 174 453, 64 570 081, 581 130 733}
���� {13, 121, 1093, 9841, 88 573, 797 161, 7 174 453, 64 570 081, 581 130 733}
����
���� {53, 485, 4373, 39 365, 354 293, 3 188 645, 28 697 813, 258 280 325, 2 324 522 933}
��
Table6 × 9n
- 1, {n, 1, 9}
��
���� {53, 485, 4373, 39 365, 354 293, 3 188 645, 28 697 813, 258 280 325, 2 324 522 933}
���� {53, 485, 4373, 39 365, 354 293, 3 188 645, 28 697 813, 258 280 325, 2 324 522 933}
����
��49

View Slide

����
�⋯
Table
��
Length
��
NestWhileListc2, 6 × 9n
- 1, # ≠ 1 &, {n, 1, 100}
% //
��
ListPlot
���� {3, 35, 9, 12, 46, 44, 30, 64, 43, 89, 152, 138, 112, 134, 132, 159, 157, 155, 153, 151,
168, 166, 164, 162, 160, 257, 156, 253, 251, 249, 247, 242, 190, 265, 263, 261,
312, 257, 255, 448, 446, 444, 406, 440, 438, 436, 434, 432, 430, 428, 414, 412,
410, 420, 418, 481, 479, 400, 410, 396, 394, 469, 467, 465, 591, 437, 435, 433,
583, 564, 579, 577, 575, 573, 571, 569, 567, 565, 563, 561, 559, 557, 555, 553,
551, 549, 547, 545, 550, 548, 539, 537, 542, 540, 538, 883, 881, 783, 781, 875}
����
20 40 60 80 100
200
400
600
800
����
�⋯
Table
��
- 1, # ≠ 1 &, {n,
��
MersennePrimeExponent[
��
Range[3]]}
���� {{3, 5, 1}, {7, 11, 17, 13, 5, 1},
{31, 47, 71, 107, 161, 121, 91, 137, 103, 155, 233, 175, 263, 395,
593, 445, 167, 251, 377, 283, 425, 319, 479, 719, 1079, 1619, 2429,
911, 1367, 2051, 3077, 577, 433, 325, 61, 23, 35, 53, 5, 1}}
����
��
Select[
��
Table[n →
�⋯
Most[
��
NestWhileList[c2, n, # ≠ 1 &]], {n, 3, 100, 2}],
��
AllTrue[
��
Last[#],
��
PrimeQ] &]
��
First /@ %
��
���� {3 → {3, 5}, 5 → {5}, 7 → {7, 11, 17, 13, 5}, 11 → {11, 17, 13, 5},
13 → {13, 5}, 17 → {17, 13, 5}, 19 → {19, 29, 11, 17, 13, 5},
29 → {29, 11, 17, 13, 5}, 37 → {37, 7, 11, 17, 13, 5},
53 → {53, 5}, 59 → {59, 89, 67, 101, 19, 29, 11, 17, 13, 5},
67 → {67, 101, 19, 29, 11, 17, 13, 5}, 89 → {89, 67, 101, 19, 29, 11, 17, 13, 5}}
���� {3, 5, 7, 11, 13, 17, 19, 29, 37, 53, 59, 67, 89}
����
50��

View Slide

����
��
First /@
��
Select[
��
Table[n →
�⋯
Most[
��
NestWhileList[c2, n, # ≠ 1 &]], {n, 3, 10 000, 2}],
��
AllTrue[
��
Last[#],
��
PrimeQ] &]
��
Prime[
��
Range[
��⋯
PrimePi[
��
Last[%]]]];
��
Complement[%, %%];
��
ListPlot[%%%]
��
ListPlot[%%%]
��
ListPlot[%%%]
���� {3, 5, 7, 11, 13, 17, 19, 29, 37, 53, 59, 67, 89, 101, 131, 149, 157,
179, 181, 197, 241, 269, 277, 349, 397, 739, 853, 1109, 1237, 1429,
1621, 1861, 1877, 2161, 2389, 2531, 2957, 3413, 3797, 4549, 5717, 7621}
����
10 20 30 40
2000
4000
6000
8000
����
200 400 600 800 1000
2000
4000
6000
8000
��51

View Slide

����
200 400 600 800
2000
4000
6000
8000
52��

View Slide

collatz-bion-experiment-2018-10-31

collatz-bion-experiment-2018-10-31

IWABUCHI Yu(u)ki

More Decks by IWABUCHI Yu(u)ki

Other Decks in Research

Featured

Transcript