Tool to test the Collatz conjecture (or Hailstone or 3n+1) and variants that divide a number by 2 if it is even and else multiply it by 3 and add 1.

Collatz Conjecture - dCode

Tag(s) : Mathematics, Fun/Miscellaneous

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Tool to test the Collatz conjecture (or Hailstone or 3n+1) and variants that divide a number by 2 if it is even and else multiply it by 3 and add 1.

Take a number number $ n $ (non-zero positive integer), if $ n $ is even, divide it by $ 2 $, else multiply by $ 3 $ and add $ 1 $. Start over with the result until you get $ 1 $.

Mathematically the algorithm is defined by the function $ f $: $$ f_{3n+1}(n) = \begin{cases}{ \frac{n}{2}} & {\text{if }}n \equiv 0 \mod{2} \\ 3n+1 & {\text{if }} n \equiv 1 \mod{2} \end{cases} $$

__Example:__ $ n=10 $, $ 10 $ is even, divide it by $ 2 $ and get $ 5 $,

$ 5 $ is odd, multiply it by $ 3 $ and add $ 1 $ to get $ 16 $,

Continue the sequence to get $ 8 $, $ 4 $, $ 2 $ and $ 1 $.

The **Collatz** conjecture stipulates that the 3n+1 algorithm will always reach the number 1.

Some numbers have surprising trajectories like 27, 255, 447, 639 or 703

No, nobody has found a number for which it does not work but nobody has found any mathematical proof that the conjecture is always true.

This is why the conjecture is also called the **Syracuse** problem or the **Collatz** problem and it is not a theorem.

A number never appears twice in the sequence.

Any sequence ends with a series of powers of 2.

An odd number is always followed by an even number.

The numbers 5 and 32 give the same result.

`// Javascript Algorithm`

function step(n) {

if (n%2 == 0) return n/2;

return 3*n+1;

}

function **collatz**(n) {

var nb = 1;

while (n != 1) {

n = step(n);

nb++;

}

return nb;

}

// Python Code Algo

def **collatz**(x):

while x != 1:

if x % 2 > 0:

x =((3 * x) + 1)

list_.append(x)

else:

x = (x / 2)

list_.append(x)

return list_

The **Collatz** conjecture/problem is also known as

- 3n + 1 conjecture

- Hailstone conjecture

- Ulam conjecture

- Kakutani's problem

- Thwaites conjecture

- Hasse's algorithm

- **Syracuse** problem

This table is for numbers until 1000 (total time => numbers)

0 | 1 |
---|---|

1 | 2 |

2 | 4 |

3 | 8 |

4 | 16 |

5 | 5, 32 |

6 | 10, 64 |

7 | 3, 20, 21, 128 |

8 | 6, 40, 42, 256 |

9 | 12, 13, 80, 84, 85, 512 |

10 | 24, 26, 160, 168, 170 |

11 | 48, 52, 53, 320, 336, 340, 341 |

12 | 17, 96, 104, 106, 113, 640, 672, 680, 682 |

13 | 34, 35, 192, 208, 212, 213, 226, 227 |

14 | 11, 68, 69, 70, 75, 384, 416, 424, 426, 452, 453, 454 |

15 | 22, 23, 136, 138, 140, 141, 150, 151, 768, 832, 848, 852, 853, 904, 906, 908, 909 |

16 | 7, 44, 45, 46, 272, 276, 277, 280, 282, 300, 301, 302 |

17 | 14, 15, 88, 90, 92, 93, 544, 552, 554, 560, 564, 565, 600, 602, 604, 605 |

18 | 28, 29, 30, 176, 180, 181, 184, 186, 201 |

19 | 9, 56, 58, 60, 61, 352, 360, 362, 368, 369, 372, 373, 401, 402, 403 |

20 | 18, 19, 112, 116, 117, 120, 122, 704, 720, 724, 725, 736, 738, 739, 744, 746, 753, 802, 803, 804, 805, 806 |

21 | 36, 37, 38, 224, 232, 234, 240, 241, 244, 245, 267 |

22 | 72, 74, 76, 77, 81, 448, 464, 468, 469, 480, 482, 483, 488, 490, 497, 534, 535, 537 |

23 | 25, 144, 148, 149, 152, 154, 162, 163, 896, 928, 936, 938, 960, 964, 965, 966, 976, 980, 981, 985, 994, 995 |

24 | 49, 50, 51, 288, 296, 298, 304, 308, 309, 321, 324, 325, 326, 331 |

25 | 98, 99, 100, 101, 102, 576, 592, 596, 597, 608, 616, 618, 625, 642, 643, 648, 650, 652, 653, 662, 663, 713, 715 |

26 | 33, 196, 197, 198, 200, 202, 204, 205, 217 |

27 | 65, 66, 67, 392, 394, 396, 397, 400, 404, 405, 408, 410, 433, 434, 435, 441, 475 |

28 | 130, 131, 132, 133, 134, 784, 788, 789, 792, 794, 800, 808, 810, 816, 820, 821, 833, 857, 866, 867, 868, 869, 870, 875, 882, 883, 950, 951, 953, 955 |

29 | 43, 260, 261, 262, 264, 266, 268, 269, 273, 289 |

30 | 86, 87, 89, 520, 522, 524, 525, 528, 529, 532, 533, 536, 538, 546, 547, 555, 571, 577, 578, 579, 583, 633, 635 |

31 | 172, 173, 174, 177, 178, 179 |

32 | 57, 59, 344, 346, 348, 349, 354, 355, 356, 357, 358, 385, 423 |

33 | 114, 115, 118, 119, 688, 692, 693, 696, 698, 705, 708, 709, 710, 712, 714, 716, 717, 729, 761, 769, 770, 771, 777, 846, 847 |

34 | 39, 228, 229, 230, 236, 237, 238 |

35 | 78, 79, 456, 458, 460, 461, 465, 472, 473, 474, 476, 477, 507, 513 |

36 | 153, 156, 157, 158, 912, 916, 917, 920, 922, 930, 931, 943, 944, 945, 946, 947, 948, 949, 952, 954, 971, 987 |

37 | 305, 306, 307, 312, 314, 315, 316, 317 |

38 | 105, 610, 611, 612, 613, 614, 624, 628, 629, 630, 631, 632, 634, 647, 683, 687 |

39 | 203, 209, 210, 211 |

40 | 406, 407, 409, 418, 419, 420, 421, 422, 431, 455 |

41 | 135, 139, 812, 813, 814, 817, 818, 819, 827, 836, 837, 838, 840, 841, 842, 843, 844, 845, 862, 863, 910, 911 |

42 | 270, 271, 278, 279, 281, 287, 303 |

43 | 540, 541, 542, 545, 551, 556, 557, 558, 561, 562, 563, 574, 575, 606, 607 |

44 | 185, 187, 191 |

45 | 361, 363, 367, 370, 371, 374, 375, 382, 383 |

46 | 123, 127, 721, 722, 723, 726, 727, 734, 735, 740, 741, 742, 747, 748, 749, 750, 764, 765, 766, 809, 891 |

47 | 246, 247, 249, 254, 255 |

48 | 481, 489, 492, 493, 494, 498, 499, 508, 509, 510, 539 |

49 | 169, 961, 962, 963, 969, 978, 979, 984, 986, 988, 989, 996, 997, 998, 999 |

50 | 329, 338, 339, 359 |

51 | 641, 657, 658, 659, 665, 676, 677, 678, 718, 719 |

52 | 219, 225, 239 |

53 | 427, 438, 439, 443, 450, 451, 478, 479 |

54 | 159, 854, 855, 876, 877, 878, 886, 887, 900, 901, 902, 907, 956, 957, 958 |

55 | 295, 318, 319 |

56 | 569, 585, 590, 591, 601, 636, 637, 638 |

58 | 379, 393, 425 |

59 | 758, 759, 767, 779, 786, 787, 801, 849, 850, 851 |

60 | 283 |

61 | 505, 511, 519, 566, 567 |

63 | 377 |

64 | 673, 679, 681, 699, 711, 754, 755 |

65 | 251 |

66 | 502, 503 |

67 | 167, 897, 905, 923 |

68 | 334, 335 |

69 | 111, 603, 615, 668, 669, 670 |

70 | 222, 223 |

71 | 444, 445, 446 |

72 | 799, 807, 888, 890, 892, 893 |

73 | 297 |

74 | 593, 594, 595 |

76 | 395 |

77 | 790, 791, 793 |

78 | 263 |

79 | 526, 527 |

80 | 175 |

81 | 350, 351 |

82 | 700, 701, 702 |

83 | 233 |

84 | 466, 467 |

85 | 155, 839, 932, 933, 934, 939 |

86 | 310, 311 |

87 | 103, 559, 620, 621, 622 |

88 | 206, 207 |

89 | 412, 413, 414 |

90 | 137, 745, 824, 826, 828, 829 |

91 | 274, 275 |

92 | 91, 548, 549, 550 |

93 | 182, 183, 993 |

94 | 364, 365, 366 |

95 | 121, 671, 728, 730, 732, 733, 743 |

96 | 242, 243 |

97 | 447, 484, 485, 486, 495 |

98 | 161, 894, 895, 968, 970, 972, 973, 977, 990, 991 |

99 | 322, 323 |

100 | 107, 644, 645, 646, 651 |

101 | 214, 215 |

102 | 71, 428, 429, 430 |

103 | 142, 143, 795, 856, 858, 860, 861 |

104 | 47, 284, 285, 286 |

105 | 94, 95, 568, 570, 572, 573 |

106 | 31, 188, 189, 190 |

107 | 62, 63, 376, 378, 380, 381 |

108 | 124, 125, 126, 752, 756, 757, 760, 762 |

109 | 41, 248, 250, 252, 253 |

110 | 82, 83, 496, 500, 501, 504, 506 |

111 | 27, 164, 165, 166, 992, 1000 |

112 | 54, 55, 328, 330, 332, 333, 337 |

113 | 108, 109, 110, 656, 660, 661, 664, 666, 674, 675 |

114 | 216, 218, 220, 221 |

115 | 73, 432, 436, 437, 440, 442, 449 |

116 | 145, 146, 147, 864, 872, 874, 880, 881, 884, 885, 898, 899, 903, 927 |

117 | 290, 291, 292, 293, 294, 299 |

118 | 97, 580, 581, 582, 584, 586, 587, 588, 589, 598, 599 |

119 | 193, 194, 195, 199 |

120 | 386, 387, 388, 389, 390, 391, 398, 399 |

121 | 129, 772, 773, 774, 776, 778, 780, 781, 782, 783, 785, 796, 797, 798 |

122 | 257, 258, 259, 265 |

123 | 514, 515, 516, 517, 518, 521, 523, 530, 531 |

124 | 171 |

125 | 342, 343, 345, 347, 353 |

126 | 684, 685, 686, 689, 690, 691, 694, 695, 697, 706, 707 |

127 | 231, 235 |

128 | 457, 459, 462, 463, 470, 471 |

129 | 913, 914, 915, 918, 919, 921, 924, 925, 926, 929, 935, 940, 941, 942, 959 |

130 | 313 |

131 | 609, 617, 619, 623, 626, 627, 639 |

133 | 411, 415, 417 |

134 | 811, 815, 822, 823, 825, 830, 831, 834, 835 |

136 | 543, 553 |

139 | 731, 737, 751 |

141 | 487, 491 |

142 | 967, 974, 975, 982, 983 |

143 | 327 |

144 | 649, 654, 655, 667 |

147 | 859, 865, 873, 879, 889 |

152 | 763, 775 |

170 | 703 |

173 | 937 |

178 | 871 |

This table shows numbers until 1000 (max number reached => numbers)

1 | 1 |
---|---|

2 | 2 |

4 | 4 |

8 | 8 |

16 | 3, 5, 6, 10, 12, 16 |

20 | 20 |

24 | 24 |

32 | 32 |

40 | 13, 26, 40 |

48 | 48 |

52 | 7, 9, 11, 14, 17, 18, 22, 28, 34, 36, 44, 52 |

56 | 56 |

64 | 21, 42, 64 |

68 | 68 |

72 | 72 |

80 | 80 |

84 | 84 |

88 | 19, 25, 29, 38, 50, 58, 76, 88 |

96 | 96 |

100 | 33, 66, 100 |

104 | 104 |

112 | 37, 74, 112 |

116 | 116 |

128 | 128 |

132 | 132 |

136 | 45, 90, 136 |

144 | 144 |

148 | 49, 98, 148 |

152 | 152 |

160 | 15, 23, 30, 35, 46, 53, 60, 70, 92, 106, 120, 140, 160 |

168 | 168 |

176 | 176 |

180 | 180 |

184 | 61, 122, 184 |

192 | 192 |

196 | 43, 57, 65, 86, 114, 130, 172, 196 |

200 | 200 |

208 | 69, 138, 208 |

212 | 212 |

224 | 224 |

228 | 228 |

232 | 51, 77, 102, 154, 204, 232 |

240 | 240 |

244 | 81, 162, 244 |

256 | 85, 170, 256 |

260 | 260 |

264 | 264 |

272 | 272 |

276 | 276 |

280 | 93, 186, 280 |

288 | 288 |

296 | 296 |

304 | 39, 59, 67, 78, 89, 101, 118, 134, 156, 178, 202, 236, 268, 304 |

308 | 308 |

312 | 312 |

320 | 320 |

324 | 324 |

336 | 336 |

340 | 75, 113, 150, 226, 300, 340 |

344 | 344 |

352 | 117, 234, 352 |

356 | 356 |

360 | 360 |

368 | 368 |

372 | 372 |

384 | 384 |

392 | 392 |

400 | 133, 266, 400 |

404 | 404 |

408 | 408 |

416 | 416 |

424 | 141, 282, 424 |

448 | 99, 149, 198, 298, 396, 448 |

452 | 452 |

456 | 456 |

464 | 464 |

468 | 468 |

472 | 157, 314, 472 |

480 | 480 |

488 | 488 |

512 | 512 |

520 | 115, 153, 173, 230, 306, 346, 460, 520 |

528 | 528 |

532 | 177, 354, 532 |

536 | 536 |

544 | 181, 362, 544 |

552 | 552 |

560 | 560 |

564 | 564 |

576 | 576 |

592 | 87, 131, 174, 197, 262, 348, 394, 524, 592 |

596 | 596 |

600 | 600 |

608 | 608 |

612 | 612 |

616 | 205, 410, 616 |

624 | 624 |

628 | 123, 139, 185, 209, 246, 278, 370, 418, 492, 556, 628 |

640 | 213, 426, 640 |

648 | 648 |

672 | 672 |

680 | 680 |

688 | 229, 458, 688 |

692 | 692 |

696 | 696 |

704 | 704 |

708 | 708 |

712 | 237, 474, 712 |

720 | 720 |

724 | 241, 482, 724 |

736 | 163, 217, 245, 326, 434, 490, 652, 736 |

740 | 740 |

744 | 744 |

768 | 768 |

784 | 261, 522, 784 |

788 | 788 |

792 | 792 |

800 | 800 |

808 | 79, 105, 119, 158, 179, 210, 238, 269, 316, 358, 420, 476, 538, 632, 716, 808 |

816 | 816 |

820 | 273, 546, 820 |

832 | 277, 554, 832 |

836 | 836 |

840 | 840 |

848 | 848 |

852 | 852 |

868 | 289, 578, 868 |

896 | 896 |

904 | 301, 602, 904 |

912 | 912 |

916 | 135, 203, 270, 305, 406, 540, 610, 812, 916 |

920 | 920 |

928 | 309, 618, 928 |

936 | 936 |

944 | 944 |

948 | 948 |

952 | 187, 211, 249, 281, 317, 374, 422, 498, 562, 634, 748, 844, 952 |

960 | 960 |

964 | 321, 642, 964 |

976 | 325, 650, 976 |

980 | 980 |

984 | 984 |

996 | 996 |

1024 | 151, 201, 227, 302, 341, 402, 454, 604, 682, 804, 908 |

1048 | 349, 698 |

1072 | 357, 714 |

1108 | 369, 738 |

1120 | 373, 746 |

1156 | 385, 770 |

1192 | 397, 794 |

1204 | 267, 401, 534, 802 |

1216 | 405, 810 |

1264 | 421, 842 |

1300 | 433, 866 |

1360 | 453, 906 |

1384 | 307, 409, 461, 614, 818, 922 |

1396 | 465, 930 |

1408 | 469, 938 |

1432 | 477, 954 |

1480 | 493, 986 |

1492 | 331, 441, 497, 662, 882, 994 |

1540 | 513 |

1576 | 525 |

1588 | 529 |

1600 | 315, 355, 473, 533, 630, 710, 946 |

1624 | 541 |

1636 | 363, 545, 726 |

1672 | 219, 247, 329, 371, 438, 494, 557, 658, 742, 876, 988 |

1684 | 561 |

1696 | 565 |

1732 | 577 |

1792 | 597 |

1816 | 403, 537, 605, 806 |

1840 | 613 |

1876 | 625 |

1888 | 279, 419, 558, 629, 838 |

1960 | 435, 653, 870 |

1972 | 657 |

2080 | 693 |

2116 | 705 |

2128 | 709 |

2152 | 717 |

2164 | 721 |

2176 | 483, 725, 966 |

2224 | 741 |

2248 | 295, 393, 443, 499, 590, 665, 749, 786, 886, 998 |

2260 | 753 |

2308 | 769 |

2368 | 789 |

2416 | 805 |

2440 | 813 |

2452 | 817 |

2464 | 547, 729, 821 |

2500 | 555, 833 |

2512 | 837 |

2536 | 375, 563, 750, 845 |

2560 | 853 |

2608 | 579, 869 |

2632 | 877 |

2728 | 909 |

2752 | 271, 361, 379, 407, 427, 481, 505, 542, 569, 611, 641, 673, 722, 758, 814, 854, 897, 917, 962 |

2836 | 945 |

2848 | 949 |

2884 | 961 |

2896 | 507, 571, 643, 761, 857, 965 |

2944 | 981 |

2968 | 439, 585, 659, 878, 989 |

2992 | 997 |

3076 | 303, 455, 606, 683, 910 |

3220 | 423, 635, 715, 846, 953 |

3256 | 723 |

3328 | 739, 985 |

3472 | 771 |

3508 | 519, 779 |

3544 | 699, 787 |

3616 | 475, 535, 633, 713, 803, 950 |

3688 | 819 |

3796 | 843 |

3904 | 867 |

3940 | 583, 777, 875 |

3976 | 883 |

4192 | 367, 489, 551, 734, 827, 931, 978 |

4264 | 631, 747, 841, 947 |

4336 | 963 |

4372 | 127, 169, 191, 225, 254, 287, 338, 339, 382, 431, 450, 451, 508, 509, 574, 601, 647, 676, 677, 678, 764, 765, 801, 862, 900, 901, 902, 971 |

4408 | 979 |

4480 | 663, 995 |

4804 | 711 |

4840 | 955 |

4912 | 727, 969 |

5128 | 759 |

5776 | 855 |

5812 | 603, 679, 905 |

5992 | 591, 887 |

6424 | 951 |

6964 | 687 |

7504 | 987 |

8080 | 559, 745, 839, 993 |

8584 | 847, 891 |

9232 | 27, 31, 41, 47, 54, 55, 62, 63, 71, 73, 82, 83, 91, 94, 95, 97, 103, 107, 108, 109, 110, 111, 121, 124, 125, 126, 129, 137, 142, 143, 145, 146, 147, 155, 159, 161, 164, 165, 166, 167, 171, 175, 182, 183, 188, 189, 190, 193, 194, 195, 199, 206, 207, 214, 215, 216, 218, 220, 221, 222, 223, 231, 233, 235, 239, 242, 243, 248, 250, 251, 252, 253, 257, 258, 259, 263, 265, 274, 275, 283, 284, 285, 286, 290, 291, 292, 293, 294, 297, 299, 310, 311, 313, 318, 319, 322, 323, 327, 328, 330, 332, 333, 334, 335, 337, 342, 343, 345, 347, 350, 351, 353, 359, 364, 365, 366, 376, 377, 378, 380, 381, 386, 387, 388, 389, 390, 391, 395, 398, 399, 411, 412, 413, 414, 415, 417, 425, 428, 429, 430, 432, 436, 437, 440, 442, 444, 445, 446, 449, 457, 459, 462, 463, 466, 467, 470, 471, 478, 479, 484, 485, 486, 487, 491, 496, 500, 501, 502, 503, 504, 506, 514, 515, 516, 517, 518, 521, 523, 526, 527, 530, 531, 539, 543, 548, 549, 550, 553, 566, 567, 568, 570, 572, 573, 580, 581, 582, 584, 586, 587, 588, 589, 593, 594, 595, 598, 599, 607, 609, 617, 619, 620, 621, 622, 623, 626, 627, 636, 637, 638, 644, 645, 646, 649, 651, 654, 655, 656, 660, 661, 664, 666, 668, 669, 670, 674, 675, 684, 685, 686, 689, 690, 691, 694, 695, 697, 700, 701, 702, 706, 707, 718, 719, 728, 730, 731, 732, 733, 737, 752, 754, 755, 756, 757, 760, 762, 763, 772, 773, 774, 775, 776, 778, 780, 781, 782, 783, 785, 790, 791, 793, 796, 797, 798, 809, 811, 815, 822, 823, 824, 825, 826, 828, 829, 830, 834, 835, 849, 850, 851, 856, 858, 859, 860, 861, 864, 865, 872, 873, 874, 880, 881, 884, 885, 888, 890, 892, 893, 898, 899, 903, 911, 913, 914, 915, 918, 919, 921, 924, 925, 926, 929, 932, 933, 934, 935, 939, 940, 941, 942, 956, 957, 958, 967, 968, 970, 972, 973, 974, 977, 982, 983, 992, 1000 |

9556 | 943 |

9880 | 975 |

10024 | 879 |

10528 | 615, 923 |

11176 | 735 |

11392 | 999 |

12148 | 799 |

13120 | 255, 383, 510, 575, 766, 863, 907 |

14308 | 495, 743, 990 |

15064 | 991 |

15856 | 927 |

18952 | 831 |

21688 | 667, 751, 889 |

39364 | 447, 511, 671, 681, 767, 795, 807, 894, 895 |

41524 | 639, 959 |

190996 | 871 |

250504 | 703, 937 |

Formulated in 1937 by Lothar **Collatz** (german mathematician), it remains unsolved: nobody has been able to prove this conjecture always ends with 1.

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