collatz-bion-experiment-2018-10-31

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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

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���� (* �� *) �� 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

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(* �� *) �� 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

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���� {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

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���� 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

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���� 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

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���� {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

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���� (* �� *) �� 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

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���� 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

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���� ⋯ 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

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