***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000826 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The stopping time of the decimal representation of the binary word for the 3x+1 problem. Prepend $1$ to the binary word, interpret it as a positive number, and count the number of halving and tripling steps needed to reach $1$. ----------------------------------------------------------------------------- References: [1] [[wikipedia:3x_+_1_problem]] ----------------------------------------------------------------------------- Code: def statistic(w): return collatz(Integer("0b1" + str(w))) @cached_function def collatz(n): if n == 1: return 0 elif is_even(n): return 1+collatz(n//2) else: return 1+collatz(3*n+1) ----------------------------------------------------------------------------- Statistic values: 0 => 1 1 => 7 00 => 2 01 => 5 10 => 8 11 => 16 000 => 3 001 => 19 010 => 6 011 => 14 100 => 9 101 => 9 110 => 17 111 => 17 0000 => 4 0001 => 12 0010 => 20 0011 => 20 0100 => 7 0101 => 7 0110 => 15 0111 => 15 1000 => 10 1001 => 23 1010 => 10 1011 => 111 1100 => 18 1101 => 18 1110 => 18 1111 => 106 00000 => 5 00001 => 26 00010 => 13 00011 => 13 00100 => 21 00101 => 21 00110 => 21 00111 => 34 01000 => 8 01001 => 109 01010 => 8 01011 => 29 01100 => 16 01101 => 16 01110 => 16 01111 => 104 10000 => 11 10001 => 24 10010 => 24 10011 => 24 10100 => 11 10101 => 11 10110 => 112 10111 => 112 11000 => 19 11001 => 32 11010 => 19 11011 => 32 11100 => 19 11101 => 19 11110 => 107 11111 => 107 000000 => 6 000001 => 27 000010 => 27 000011 => 27 000100 => 14 000101 => 14 000110 => 14 000111 => 102 001000 => 22 001001 => 115 001010 => 22 001011 => 14 001100 => 22 001101 => 22 001110 => 35 001111 => 35 010000 => 9 010001 => 22 010010 => 110 010011 => 110 010100 => 9 010101 => 9 010110 => 30 010111 => 30 011000 => 17 011001 => 30 011010 => 17 011011 => 92 011100 => 17 011101 => 17 011110 => 105 011111 => 105 100000 => 12 100001 => 118 100010 => 25 100011 => 25 100100 => 25 100101 => 25 100110 => 25 100111 => 87 101000 => 12 101001 => 38 101010 => 12 101011 => 100 101100 => 113 101101 => 113 101110 => 113 101111 => 69 110000 => 20 110001 => 12 110010 => 33 110011 => 33 110100 => 20 110101 => 20 110110 => 33 110111 => 33 111000 => 20 111001 => 95 111010 => 20 111011 => 46 111100 => 108 111101 => 108 111110 => 108 111111 => 46 0000000 => 7 0000001 => 121 0000010 => 28 0000011 => 28 0000100 => 28 0000101 => 28 0000110 => 28 0000111 => 41 0001000 => 15 0001001 => 90 0001010 => 15 0001011 => 41 0001100 => 15 0001101 => 15 0001110 => 103 0001111 => 103 0010000 => 23 0010001 => 116 0010010 => 116 0010011 => 116 0010100 => 23 0010101 => 23 0010110 => 15 0010111 => 15 0011000 => 23 0011001 => 36 0011010 => 23 0011011 => 85 0011100 => 36 0011101 => 36 0011110 => 36 0011111 => 54 0100000 => 10 0100001 => 98 0100010 => 23 0100011 => 23 0100100 => 111 0100101 => 111 0100110 => 111 0100111 => 67 0101000 => 10 0101001 => 49 0101010 => 10 0101011 => 124 0101100 => 31 0101101 => 31 0101110 => 31 0101111 => 80 0110000 => 18 0110001 => 31 0110010 => 31 0110011 => 31 0110100 => 18 0110101 => 18 0110110 => 93 0110111 => 93 0111000 => 18 0111001 => 44 0111010 => 18 0111011 => 44 0111100 => 106 0111101 => 106 0111110 => 106 0111111 => 44 1000000 => 13 1000001 => 119 1000010 => 119 1000011 => 119 1000100 => 26 1000101 => 26 1000110 => 26 1000111 => 119 1001000 => 26 1001001 => 18 1001010 => 26 1001011 => 39 1001100 => 26 1001101 => 26 1001110 => 88 1001111 => 88 1010000 => 13 1010001 => 39 1010010 => 39 1010011 => 39 1010100 => 13 1010101 => 13 1010110 => 101 1010111 => 101 1011000 => 114 1011001 => 26 1011010 => 114 1011011 => 52 1011100 => 114 1011101 => 114 1011110 => 70 1011111 => 70 1100000 => 21 1100001 => 52 1100010 => 13 1100011 => 13 1100100 => 34 1100101 => 34 1100110 => 34 1100111 => 127 1101000 => 21 1101001 => 83 1101010 => 21 1101011 => 127 1101100 => 34 1101101 => 34 1101110 => 34 1101111 => 52 1110000 => 21 1110001 => 21 1110010 => 96 1110011 => 96 1110100 => 21 1110101 => 21 1110110 => 47 1110111 => 47 1111000 => 109 1111001 => 47 1111010 => 109 1111011 => 65 1111100 => 109 1111101 => 109 1111110 => 47 1111111 => 47 00000000 => 8 00000001 => 122 00000010 => 122 00000011 => 122 00000100 => 29 00000101 => 29 00000110 => 29 00000111 => 78 00001000 => 29 00001001 => 122 00001010 => 29 00001011 => 21 00001100 => 29 00001101 => 29 00001110 => 42 00001111 => 42 00010000 => 16 00010001 => 29 00010010 => 91 00010011 => 91 00010100 => 16 00010101 => 16 00010110 => 42 00010111 => 42 00011000 => 16 00011001 => 42 00011010 => 16 00011011 => 60 00011100 => 104 00011101 => 104 00011110 => 104 00011111 => 42 00100000 => 24 00100001 => 29 00100010 => 117 00100011 => 117 00100100 => 117 00100101 => 117 00100110 => 117 00100111 => 55 00101000 => 24 00101001 => 73 00101010 => 24 00101011 => 117 00101100 => 16 00101101 => 16 00101110 => 16 00101111 => 42 00110000 => 24 00110001 => 37 00110010 => 37 00110011 => 37 00110100 => 24 00110101 => 24 00110110 => 86 00110111 => 86 00111000 => 37 00111001 => 130 00111010 => 37 00111011 => 37 00111100 => 37 00111101 => 37 00111110 => 55 00111111 => 55 01000000 => 11 01000001 => 24 01000010 => 99 01000011 => 99 01000100 => 24 01000101 => 24 01000110 => 24 01000111 => 143 01001000 => 112 01001001 => 50 01001010 => 112 01001011 => 24 01001100 => 112 01001101 => 112 01001110 => 68 01001111 => 68 01010000 => 11 01010001 => 112 01010010 => 50 01010011 => 50 01010100 => 11 01010101 => 11 01010110 => 125 01010111 => 125 01011000 => 32 01011001 => 125 01011010 => 32 01011011 => 125 01011100 => 32 01011101 => 32 01011110 => 81 01011111 => 81 01100000 => 19 01100001 => 125 01100010 => 32 01100011 => 32 01100100 => 32 01100101 => 32 01100110 => 32 01100111 => 50 01101000 => 19 01101001 => 45 01101010 => 19 01101011 => 45 01101100 => 94 01101101 => 94 01101110 => 94 01101111 => 45 01110000 => 19 01110001 => 19 01110010 => 45 01110011 => 45 01110100 => 19 01110101 => 19 01110110 => 45 01110111 => 45 01111000 => 107 01111001 => 63 01111010 => 107 01111011 => 58 01111100 => 107 01111101 => 107 01111110 => 45 01111111 => 45 10000000 => 14 10000001 => 32 10000010 => 120 10000011 => 120 10000100 => 120 10000101 => 120 10000110 => 120 10000111 => 120 10001000 => 27 10001001 => 58 10001010 => 27 10001011 => 76 10001100 => 27 10001101 => 27 10001110 => 120 10001111 => 120 10010000 => 27 10010001 => 19 10010010 => 19 10010011 => 19 10010100 => 27 10010101 => 27 10010110 => 40 10010111 => 40 10011000 => 27 10011001 => 40 10011010 => 27 10011011 => 133 10011100 => 89 10011101 => 89 10011110 => 89 10011111 => 133 10100000 => 14 10100001 => 133 10100010 => 40 10100011 => 40 10100100 => 40 10100101 => 40 10100110 => 40 10100111 => 32 10101000 => 14 10101001 => 58 10101010 => 14 10101011 => 53 10101100 => 102 10101101 => 102 10101110 => 102 10101111 => 40 10110000 => 115 10110001 => 27 10110010 => 27 10110011 => 27 10110100 => 115 10110101 => 115 10110110 => 53 10110111 => 53 10111000 => 115 10111001 => 27 10111010 => 115 10111011 => 53 10111100 => 71 10111101 => 71 10111110 => 71 10111111 => 97 11000000 => 22 11000001 => 115 11000010 => 53 11000011 => 53 11000100 => 14 11000101 => 14 11000110 => 14 11000111 => 40 11001000 => 35 11001001 => 128 11001010 => 35 11001011 => 128 11001100 => 35 11001101 => 35 11001110 => 128 11001111 => 128 11010000 => 22 11010001 => 35 11010010 => 84 11010011 => 84 11010100 => 22 11010101 => 22 11010110 => 128 11010111 => 128 11011000 => 35 11011001 => 35 11011010 => 35 11011011 => 27 11011100 => 35 11011101 => 35 11011110 => 53 11011111 => 53 11100000 => 22 11100001 => 48 11100010 => 22 11100011 => 22 11100100 => 97 11100101 => 97 11100110 => 97 11100111 => 141 11101000 => 22 11101001 => 48 11101010 => 22 11101011 => 141 11101100 => 48 11101101 => 48 11101110 => 48 11101111 => 97 11110000 => 110 11110001 => 22 11110010 => 48 11110011 => 48 11110100 => 110 11110101 => 110 11110110 => 66 11110111 => 66 11111000 => 110 11111001 => 61 11111010 => 110 11111011 => 35 11111100 => 48 11111101 => 48 11111110 => 48 11111111 => 61 000000000 => 9 000000001 => 35 000000010 => 123 000000011 => 123 000000100 => 123 000000101 => 123 000000110 => 123 000000111 => 61 000001000 => 30 000001001 => 123 000001010 => 30 000001011 => 123 000001100 => 30 000001101 => 30 000001110 => 79 000001111 => 79 000010000 => 30 000010001 => 30 000010010 => 123 000010011 => 123 000010100 => 30 000010101 => 30 000010110 => 22 000010111 => 22 000011000 => 30 000011001 => 22 000011010 => 30 000011011 => 48 000011100 => 43 000011101 => 43 000011110 => 43 000011111 => 136 000100000 => 17 000100001 => 43 000100010 => 30 000100011 => 30 000100100 => 92 000100101 => 92 000100110 => 92 000100111 => 43 000101000 => 17 000101001 => 136 000101010 => 17 000101011 => 30 000101100 => 43 000101101 => 43 000101110 => 43 000101111 => 87 000110000 => 17 000110001 => 43 000110010 => 43 000110011 => 43 000110100 => 17 000110101 => 17 000110110 => 61 000110111 => 61 000111000 => 105 000111001 => 56 000111010 => 105 000111011 => 30 000111100 => 105 000111101 => 105 000111110 => 43 000111111 => 43 001000000 => 25 001000001 => 30 001000010 => 30 001000011 => 30 001000100 => 118 001000101 => 118 001000110 => 118 001000111 => 30 001001000 => 118 001001001 => 56 001001010 => 118 001001011 => 118 001001100 => 118 001001101 => 118 001001110 => 56 001001111 => 56 001010000 => 25 001010001 => 74 001010010 => 74 001010011 => 74 001010100 => 25 001010101 => 25 001010110 => 118 001010111 => 118 001011000 => 17 001011001 => 56 001011010 => 17 001011011 => 69 001011100 => 17 001011101 => 17 001011110 => 43 001011111 => 43 001100000 => 25 001100001 => 131 001100010 => 38 001100011 => 38 001100100 => 38 001100101 => 38 001100110 => 38 001100111 => 69 001101000 => 25 001101001 => 131 001101010 => 25 001101011 => 131 001101100 => 87 001101101 => 87 001101110 => 87 001101111 => 131 001110000 => 38 001110001 => 25 001110010 => 131 001110011 => 131 001110100 => 38 001110101 => 38 001110110 => 38 001110111 => 38 001111000 => 38 001111001 => 30 001111010 => 38 001111011 => 30 001111100 => 56 001111101 => 56 001111110 => 56 001111111 => 131 010000000 => 12 010000001 => 51 010000010 => 25 010000011 => 25 010000100 => 100 010000101 => 100 010000110 => 100 010000111 => 38 010001000 => 25 010001001 => 144 010001010 => 25 010001011 => 100 010001100 => 25 010001101 => 25 010001110 => 144 010001111 => 144 010010000 => 113 010010001 => 51 010010010 => 51 010010011 => 51 010010100 => 113 010010101 => 113 010010110 => 25 010010111 => 25 010011000 => 113 010011001 => 51 010011010 => 113 010011011 => 144 010011100 => 69 010011101 => 69 010011110 => 69 010011111 => 95 010100000 => 12 010100001 => 64 010100010 => 113 010100011 => 113 010100100 => 51 010100101 => 51 010100110 => 51 010100111 => 64 010101000 => 12 010101001 => 64 010101010 => 12 010101011 => 38 010101100 => 126 010101101 => 126 010101110 => 126 010101111 => 38 010110000 => 33 010110001 => 126 010110010 => 126 010110011 => 126 010110100 => 33 010110101 => 33 010110110 => 126 010110111 => 126 010111000 => 33 010111001 => 126 010111010 => 33 010111011 => 64 010111100 => 82 010111101 => 82 010111110 => 82 010111111 => 170 011000000 => 20 011000001 => 33 011000010 => 126 011000011 => 126 011000100 => 33 011000101 => 33 011000110 => 33 011000111 => 64 011001000 => 33 011001001 => 25 011001010 => 33 011001011 => 25 011001100 => 33 011001101 => 33 011001110 => 51 011001111 => 51 011010000 => 20 011010001 => 46 011010010 => 46 011010011 => 46 011010100 => 20 011010101 => 20 011010110 => 46 011010111 => 46 011011000 => 95 011011001 => 33 011011010 => 95 011011011 => 139 011011100 => 95 011011101 => 95 011011110 => 46 011011111 => 46 011100000 => 20 011100001 => 139 011100010 => 20 011100011 => 20 011100100 => 46 011100101 => 46 011100110 => 46 011100111 => 95 011101000 => 20 011101001 => 90 011101010 => 20 011101011 => 46 011101100 => 46 011101101 => 46 011101110 => 46 011101111 => 139 011110000 => 108 011110001 => 20 011110010 => 64 011110011 => 64 011110100 => 108 011110101 => 108 011110110 => 59 011110111 => 59 011111000 => 108 011111001 => 33 011111010 => 108 011111011 => 152 011111100 => 46 011111101 => 46 011111110 => 46 011111111 => 59 100000000 => 15 100000001 => 33 100000010 => 33 100000011 => 33 100000100 => 121 100000101 => 121 100000110 => 121 100000111 => 152 100001000 => 121 100001001 => 33 100001010 => 121 100001011 => 59 100001100 => 121 100001101 => 121 100001110 => 121 100001111 => 121 100010000 => 28 100010001 => 121 100010010 => 59 100010011 => 59 100010100 => 28 100010101 => 28 100010110 => 77 100010111 => 77 100011000 => 28 100011001 => 77 100011010 => 28 100011011 => 103 100011100 => 121 100011101 => 121 100011110 => 121 100011111 => 72 100100000 => 28 100100001 => 59 100100010 => 20 100100011 => 20 100100100 => 20 100100101 => 20 100100110 => 20 100100111 => 72 100101000 => 28 100101001 => 46 100101010 => 28 100101011 => 134 100101100 => 41 100101101 => 41 100101110 => 41 100101111 => 134 100110000 => 28 100110001 => 41 100110010 => 41 100110011 => 41 100110100 => 28 100110101 => 28 100110110 => 134 100110111 => 134 100111000 => 90 100111001 => 134 100111010 => 90 100111011 => 41 100111100 => 90 100111101 => 90 100111110 => 134 100111111 => 134 101000000 => 15 101000001 => 28 101000010 => 134 101000011 => 134 101000100 => 41 101000101 => 41 101000110 => 41 101000111 => 85 101001000 => 41 101001001 => 41 101001010 => 41 101001011 => 41 101001100 => 41 101001101 => 41 101001110 => 33 101001111 => 33 101010000 => 15 101010001 => 59 101010010 => 59 101010011 => 59 101010100 => 15 101010101 => 15 101010110 => 54 101010111 => 54 101011000 => 103 101011001 => 28 101011010 => 103 101011011 => 147 101011100 => 103 101011101 => 103 101011110 => 41 101011111 => 41 101100000 => 116 101100001 => 147 101100010 => 28 101100011 => 28 101100100 => 28 101100101 => 28 101100110 => 28 101100111 => 178 101101000 => 116 101101001 => 147 101101010 => 116 101101011 => 28 101101100 => 54 101101101 => 54 101101110 => 54 101101111 => 147 101110000 => 116 101110001 => 116 101110010 => 28 101110011 => 28 101110100 => 116 101110101 => 116 101110110 => 54 101110111 => 54 101111000 => 72 101111001 => 147 101111010 => 72 101111011 => 46 101111100 => 72 101111101 => 72 101111110 => 98 101111111 => 98 110000000 => 23 110000001 => 67 110000010 => 116 110000011 => 116 110000100 => 54 110000101 => 54 110000110 => 54 110000111 => 116 110001000 => 15 110001001 => 67 110001010 => 15 110001011 => 54 110001100 => 15 110001101 => 15 110001110 => 41 110001111 => 41 110010000 => 36 110010001 => 129 110010010 => 129 110010011 => 129 110010100 => 36 110010101 => 36 110010110 => 129 110010111 => 129 110011000 => 36 110011001 => 129 110011010 => 36 110011011 => 67 110011100 => 129 110011101 => 129 110011110 => 129 110011111 => 116 110100000 => 23 110100001 => 129 110100010 => 36 110100011 => 36 110100100 => 85 110100101 => 85 110100110 => 85 110100111 => 129 110101000 => 23 110101001 => 173 110101010 => 23 110101011 => 85 110101100 => 129 110101101 => 129 110101110 => 129 110101111 => 36 110110000 => 36 110110001 => 36 110110010 => 36 110110011 => 36 110110100 => 36 110110101 => 36 110110110 => 28 110110111 => 28 110111000 => 36 110111001 => 28 110111010 => 36 110111011 => 28 110111100 => 54 110111101 => 54 110111110 => 54 110111111 => 129 111000000 => 23 111000001 => 49 111000010 => 49 111000011 => 49 111000100 => 23 111000101 => 23 111000110 => 23 111000111 => 142 111001000 => 98 111001001 => 49 111001010 => 98 111001011 => 36 111001100 => 98 111001101 => 98 111001110 => 142 111001111 => 142 111010000 => 23 111010001 => 98 111010010 => 49 111010011 => 49 111010100 => 23 111010101 => 23 111010110 => 142 111010111 => 142 111011000 => 49 111011001 => 23 111011010 => 49 111011011 => 36 111011100 => 49 111011101 => 49 111011110 => 98 111011111 => 98 111100000 => 111 111100001 => 93 111100010 => 23 111100011 => 23 111100100 => 49 111100101 => 49 111100110 => 49 111100111 => 49 111101000 => 111 111101001 => 142 111101010 => 111 111101011 => 41 111101100 => 67 111101101 => 67 111101110 => 67 111101111 => 93 111110000 => 111 111110001 => 111 111110010 => 62 111110011 => 62 111110100 => 111 111110101 => 111 111110110 => 36 111110111 => 36 111111000 => 49 111111001 => 155 111111010 => 49 111111011 => 62 111111100 => 49 111111101 => 49 111111110 => 62 111111111 => 62 0000000001 => 36 0010000001 => 150 0010101101 => 119 0010101011 => 119 0010100111 => 49 0010011101 => 57 0010011011 => 57 0010010111 => 119 0010001111 => 31 0001110101 => 106 0001110011 => 57 0001101101 => 62 0001101011 => 36 0001100111 => 137 0001011101 => 44 0001011011 => 93 0001010111 => 31 0001001111 => 44 0000111101 => 44 0000111011 => 137 0000110111 => 49 0000101111 => 168 0000011111 => 168 0000000000 => 10 0000000110 => 124 0000011000 => 31 0000011110 => 80 0000010010 => 124 0001100000 => 18 0001100110 => 44 0001111000 => 106 0001111110 => 44 0001110010 => 57 0001001000 => 93 0001001110 => 44 0001000010 => 44 0001011010 => 44 0010101010 => 26 0010101000 => 26 0010101100 => 119 0010100100 => 75 0010010100 => 119 0001010100 => 18 0010101110 => 119 0010100000 => 26 0010000100 => 31 0000010100 => 31 0000000010 => 36 0000001110 => 62 0000111110 => 137 0000000100 => 124 0000001010 => 124 0000010110 => 124 0000101110 => 23 0001011110 => 88 0001101110 => 62 0001110110 => 31 0001111010 => 106 0001111100 => 44 0000000101 => 124 0000001101 => 124 0000011101 => 80 0001111101 => 44 0000001001 => 155 0000010101 => 31 0000101101 => 23 0010110001 => 57 0001100001 => 44 0001000001 => 44 0000111111 => 137 0001110100 => 106 0001101100 => 62 0010011100 => 57 0001011100 => 44 0000111100 => 44 0001101010 => 18 0010011010 => 119 0000111010 => 44 0010100110 => 75 0010010110 => 119 0001010110 => 31 0000110110 => 49 0010001110 => 31 0001101001 => 36 0000110001 => 23 0000010001 => 124 0000001100 => 124 0000011001 => 62 0000110011 => 23 0000110100 => 31 0000000011 => 36 0000011011 => 93 0001101111 => 62 0000110101 => 31 0010000010 => 31 0001000011 => 44 0010011110 => 57 0001010000 => 18 0001000100 => 31 0001010101 => 18 0001001011 => 31 0001100011 => 44 0001110111 => 31 0010000000 => 26 0000000111 => 36 0000001111 => 62 0001111111 => 57 0000001011 => 124 0000010111 => 124 0001011111 => 88 0001111001 => 150 0000011010 => 31 0000101010 => 31 0000100101 => 124 0010010001 => 57 0010000101 => 31 0010010101 => 119 0010010011 => 57 0000101011 => 62 0001000111 => 75 0000100111 => 75 0010010010 => 57 0001001010 => 93 0001001001 => 137 0001000101 => 31 0000100011 => 31 0000100100 => 124 0000100010 => 31 0010110000 => 18 0000011100 => 80 0000001000 => 124 0000101100 => 23 0010101111 => 70 0000101000 => 31 0010001000 => 119 0001001100 => 93 0001000110 => 31 0001100100 => 44 0001100010 => 44 0000100110 => 124 0000110010 => 23 0010010000 => 119 0001010010 => 137 0001000000 => 18 0001011001 => 137 0001100101 => 44 0001110001 => 18 0010011001 => 31 0010100101 => 75 0010101001 => 101 0010011111 => 88 0000111001 => 137 0010001101 => 119 0001001101 => 93 0001010001 => 93 0001010011 => 137 0001111011 => 57 0010011000 => 119 0001011000 => 44 0000100000 => 31 0010001011 => 150 0010000111 => 57 0000010011 => 124 0010001100 => 119 0010000110 => 31 0000110000 => 31 0000100001 => 80 0010100011 => 75 0010100001 => 150 0000111000 => 44 0001101000 => 18 0000010000 => 31 0000101001 => 75 0010100010 => 75 0010001001 => 181 0010000011 => 31 0010001010 => 119 0001110000 => 106 ----------------------------------------------------------------------------- Created: May 30, 2017 at 22:00 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 30, 2017 at 22:00 by Martin Rubey