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

���� ⋯ 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�� collatz-bion-experiment-2018-10-31.nb

���� ⋯ 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 collatz-bion-experiment-2018-10-31.nb ��3

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

�� 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 collatz-bion-experiment-2018-10-31.nb ��5

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

���� 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 collatz-bion-experiment-2018-10-31.nb ��7

���� (* �� *) 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�� collatz-bion-experiment-2018-10-31.nb

���� (* �� *) �� 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 collatz-bion-experiment-2018-10-31.nb ��9

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

���� 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 collatz-bion-experiment-2018-10-31.nb ��11

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

���� 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 collatz-bion-experiment-2018-10-31.nb ��13

���� (* �� *) �� 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�� collatz-bion-experiment-2018-10-31.nb

���� 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 collatz-bion-experiment-2018-10-31.nb ��15

���� (* 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�� collatz-bion-experiment-2018-10-31.nb

(* �� *) �� 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

���� {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[ �� NestWhileList[c2, n, # ≠ 1 &]], {n, 1, 250, 2}]; �� First /@ �� Select[%, �� Last[#] ⩵ 1 &] ���� {1} ���� Table[n → �� Length[ �� NestWhileList[c2, n, # ≠ 1 &]], {n, 1, 350, 2}]; �� First /@ �� Select[%, �� Last[#] ⩵ 2 &] ���� {5, 21, 85, 341} ���� Table[n → �� Length[ �� NestWhileList[c2, n, # ≠ 1 &]], {n, 1, 250, 2}]; �� First /@ �� Select[%, �� Last[#] ⩵ 10 &] ���� {43, 87, 89, 173, 177, 179} 18�� collatz-bion-experiment-2018-10-31.nb

(* �� *) ��⋯ 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 = �� 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

���� 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

���� 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

���� (* �� *) �� 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, 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

���� 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}}] & /@ �� 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 collatz-bion-experiment-2018-10-31.nb ��23

���� ⋯ Table �� 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, 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 �� NestWhileList[c2, 3077, # ≠ 1 &] // �� Length ���� 10 ���� 10 ���� Floor2 �� Log23 × 449 × 28 - 2 �� NestWhileList[c2, 449, # ≠ 1 &] // �� Length ���� 36 ���� 42 24�� collatz-bion-experiment-2018-10-31.nb

���� Table �� Numerator /@  �� 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 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[ �� NestWhileList[c2, n, # ≠ 1 &]], 1]}, 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  collatz-bion-experiment-2018-10-31.nb ��25

���� ⋯ TreePlot �� Union �� Flatten ��⋯ Table ��
Rule @@ # & /@ ��⋯ Partition[ �� Reverse[2^ �� With[{li = �� Mod[ �⋯ Log2[ �� NestWhileList[c2, n, # ≠ 1 &]], 1]}, 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�� collatz-bion-experiment-2018-10-31.nb

���� 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 collatz-bion-experiment-2018-10-31.nb ��27

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

���� 47 71 91 121 485 95 103 137 107
161 143 155 167 445 175 233 215 251 263 283 377 323 395 593 collatz-bion-experiment-2018-10-31.nb ��29

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

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

���� {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�� collatz-bion-experiment-2018-10-31.nb

���� 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] collatz-bion-experiment-2018-10-31.nb ��33

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

���� 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} collatz-bion-experiment-2018-10-31.nb ��35

���� 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 ���� Hyperlink["https://oeis.org/A007310"] �� Select[ �� Range[203], �� MemberQ[{1, 5}, �� Mod[#, 6]] &] ���� https://oeis.org/A007310 ���� {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�� collatz-bion-experiment-2018-10-31.nb

���� c2 /@ �� Range[2, 203] �� ListPlot[%] �� Hyperlink["https://oeis.org/A139391"]
���� {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 ���� https://oeis.org/A139391 ���� {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} collatz-bion-experiment-2018-10-31.nb ��37

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

���� 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 collatz-bion-experiment-2018-10-31.nb ��39

���� 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} �� FindSequenceFunction[ %, n] ���� {10, 15, 26, 31, 42, 47} ���� 1 2 9 - 3 (-1)n - 8 (-1)2 n + 16 n 40�� collatz-bion-experiment-2018-10-31.nb

���� 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 collatz-bion-experiment-2018-10-31.nb ��41

���� {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} �� FindSequenceFunction[ %, 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�� collatz-bion-experiment-2018-10-31.nb

���� 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

���� {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�� collatz-bion-experiment-2018-10-31.nb

���� (* �� *) �� 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 �� Hyperlink["https://oeis.org/A075486"] ���� {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 ���� https://oeis.org/A075486 collatz-bion-experiment-2018-10-31.nb ��45

���� 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 �� Hyperlink["https://oeis.org/A075485"] ���� {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 ���� https://oeis.org/A075485 ���� Length /@ �⋯ Table �� NestWhileListc2, 2n - 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�� collatz-bion-experiment-2018-10-31.nb

���� 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 = �� 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, 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 collatz-bion-experiment-2018-10-31.nb ��47

���� 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  �� NestWhileListc2, 22 n - 1, # ≠ 1 &, �� NestWhileListc2, 22 n-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�� collatz-bion-experiment-2018-10-31.nb

���� ⋯ Table �� Last �� Select  �� NestWhileListc2,
22 n - 1, # ≠ 1 &, �� NestWhileListc2, 22 n-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} �� Hyperlink["https://oeis.org/A096053"] ���� {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} ���� https://oeis.org/A096053 ���� {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} �� Hyperlink["https://oeis.org/A198963"] ���� {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} ���� https://oeis.org/A198963 collatz-bion-experiment-2018-10-31.nb ��49

���� ⋯ 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 �� NestWhileListc2, 2n - 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 /@ % �� Hyperlink["https://oeis.org/A177000"] ���� {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} ���� https://oeis.org/A177000 50�� collatz-bion-experiment-2018-10-31.nb

���� 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 collatz-bion-experiment-2018-10-31.nb ��51

���� 200 400 600 800 2000 4000 6000 8000 52��
collatz-bion-experiment-2018-10-31.nb

collatz-bion-experiment-2018-10-31

collatz-bion-experiment-2018-10-31

More Decks by IWABUCHI Yu(u)ki butchi

Other Decks in Research

Featured

Transcript