***************************************************************************** * 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: St001985 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The sum of the greatest common divisors of all subsets of the parts of an integer partition. ----------------------------------------------------------------------------- References: [1] Malikiosis, R. D., Santos, F., Schymura, M. Linearly exponential checking is enough for the lonely runner conjecture and some of its variants [[MathSciNet:4968011]] ----------------------------------------------------------------------------- Code: def statistic(pi): return sum(gcd(S) for S in subsets(pi)) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [2] => 2 [1,1] => 3 [3] => 3 [2,1] => 4 [1,1,1] => 7 [4] => 4 [3,1] => 5 [2,2] => 6 [2,1,1] => 8 [1,1,1,1] => 15 [5] => 5 [4,1] => 6 [3,2] => 6 [3,1,1] => 9 [2,2,1] => 10 [2,1,1,1] => 16 [1,1,1,1,1] => 31 [6] => 6 [5,1] => 7 [4,2] => 8 [4,1,1] => 10 [3,3] => 9 [3,2,1] => 10 [3,1,1,1] => 17 [2,2,2] => 14 [2,2,1,1] => 18 [2,1,1,1,1] => 32 [1,1,1,1,1,1] => 63 [7] => 7 [6,1] => 8 [5,2] => 8 [5,1,1] => 11 [4,3] => 8 [4,2,1] => 12 [4,1,1,1] => 18 [3,3,1] => 13 [3,2,2] => 12 [3,2,1,1] => 18 [3,1,1,1,1] => 33 [2,2,2,1] => 22 [2,2,1,1,1] => 34 [2,1,1,1,1,1] => 64 [1,1,1,1,1,1,1] => 127 [8] => 8 [7,1] => 9 [6,2] => 10 [6,1,1] => 12 [5,3] => 9 [5,2,1] => 12 [5,1,1,1] => 19 [4,4] => 12 [4,3,1] => 12 [4,2,2] => 16 [4,2,1,1] => 20 [4,1,1,1,1] => 34 [3,3,2] => 14 [3,3,1,1] => 21 [3,2,2,1] => 20 [3,2,1,1,1] => 34 [3,1,1,1,1,1] => 65 [2,2,2,2] => 30 [2,2,2,1,1] => 38 [2,2,1,1,1,1] => 66 [2,1,1,1,1,1,1] => 128 [1,1,1,1,1,1,1,1] => 255 [9] => 9 [8,1] => 10 [7,2] => 10 [7,1,1] => 13 [6,3] => 12 [6,2,1] => 14 [6,1,1,1] => 20 [5,4] => 10 [5,3,1] => 13 [5,2,2] => 14 [5,2,1,1] => 20 [5,1,1,1,1] => 35 [4,4,1] => 16 [4,3,2] => 14 [4,3,1,1] => 20 [4,2,2,1] => 24 [4,2,1,1,1] => 36 [4,1,1,1,1,1] => 66 [3,3,3] => 21 [3,3,2,1] => 22 [3,3,1,1,1] => 37 [3,2,2,2] => 24 [3,2,2,1,1] => 36 [3,2,1,1,1,1] => 66 [3,1,1,1,1,1,1] => 129 [2,2,2,2,1] => 46 [2,2,2,1,1,1] => 70 [2,2,1,1,1,1,1] => 130 [2,1,1,1,1,1,1,1] => 256 [1,1,1,1,1,1,1,1,1] => 511 [10] => 10 [9,1] => 11 [8,2] => 12 [8,1,1] => 14 [7,3] => 11 [7,2,1] => 14 [7,1,1,1] => 21 [6,4] => 12 [6,3,1] => 16 [6,2,2] => 18 [6,2,1,1] => 22 [6,1,1,1,1] => 36 [5,5] => 15 [5,4,1] => 14 [5,3,2] => 14 [5,3,1,1] => 21 [5,2,2,1] => 22 [5,2,1,1,1] => 36 [5,1,1,1,1,1] => 67 [4,4,2] => 20 [4,4,1,1] => 24 [4,3,3] => 16 [4,3,2,1] => 22 [4,3,1,1,1] => 36 [4,2,2,2] => 32 [4,2,2,1,1] => 40 [4,2,1,1,1,1] => 68 [4,1,1,1,1,1,1] => 130 [3,3,3,1] => 29 [3,3,2,2] => 24 [3,3,2,1,1] => 38 [3,3,1,1,1,1] => 69 [3,2,2,2,1] => 40 [3,2,2,1,1,1] => 68 [3,2,1,1,1,1,1] => 130 [3,1,1,1,1,1,1,1] => 257 [2,2,2,2,2] => 62 [2,2,2,2,1,1] => 78 [2,2,2,1,1,1,1] => 134 [2,2,1,1,1,1,1,1] => 258 [2,1,1,1,1,1,1,1,1] => 512 [1,1,1,1,1,1,1,1,1,1] => 1023 [11] => 11 [10,1] => 12 [9,2] => 12 [9,1,1] => 15 [8,3] => 12 [8,2,1] => 16 [8,1,1,1] => 22 [7,4] => 12 [7,3,1] => 15 [7,2,2] => 16 [7,2,1,1] => 22 [7,1,1,1,1] => 37 [6,5] => 12 [6,4,1] => 16 [6,3,2] => 18 [6,3,1,1] => 24 [6,2,2,1] => 26 [6,2,1,1,1] => 38 [6,1,1,1,1,1] => 68 [5,5,1] => 19 [5,4,2] => 16 [5,4,1,1] => 22 [5,3,3] => 17 [5,3,2,1] => 22 [5,3,1,1,1] => 37 [5,2,2,2] => 26 [5,2,2,1,1] => 38 [5,2,1,1,1,1] => 68 [5,1,1,1,1,1,1] => 131 [4,4,3] => 18 [4,4,2,1] => 28 [4,4,1,1,1] => 40 [4,3,3,1] => 24 [4,3,2,2] => 26 [4,3,2,1,1] => 38 [4,3,1,1,1,1] => 68 [4,2,2,2,1] => 48 [4,2,2,1,1,1] => 72 [4,2,1,1,1,1,1] => 132 [4,1,1,1,1,1,1,1] => 258 [3,3,3,2] => 30 [3,3,3,1,1] => 45 [3,3,2,2,1] => 40 [3,3,2,1,1,1] => 70 [3,3,1,1,1,1,1] => 133 [3,2,2,2,2] => 48 [3,2,2,2,1,1] => 72 [3,2,2,1,1,1,1] => 132 [3,2,1,1,1,1,1,1] => 258 [3,1,1,1,1,1,1,1,1] => 513 [2,2,2,2,2,1] => 94 [2,2,2,2,1,1,1] => 142 [2,2,2,1,1,1,1,1] => 262 [2,2,1,1,1,1,1,1,1] => 514 [2,1,1,1,1,1,1,1,1,1] => 1024 [1,1,1,1,1,1,1,1,1,1,1] => 2047 [12] => 12 [11,1] => 13 [10,2] => 14 [10,1,1] => 16 [9,3] => 15 [9,2,1] => 16 [9,1,1,1] => 23 [8,4] => 16 [8,3,1] => 16 [8,2,2] => 20 [8,2,1,1] => 24 [8,1,1,1,1] => 38 [7,5] => 13 [7,4,1] => 16 [7,3,2] => 16 [7,3,1,1] => 23 [7,2,2,1] => 24 [7,2,1,1,1] => 38 [7,1,1,1,1,1] => 69 [6,6] => 18 [6,5,1] => 16 [6,4,2] => 20 [6,4,1,1] => 24 [6,3,3] => 24 [6,3,2,1] => 26 [6,3,1,1,1] => 40 [6,2,2,2] => 34 [6,2,2,1,1] => 42 [6,2,1,1,1,1] => 70 [6,1,1,1,1,1,1] => 132 [5,5,2] => 20 [5,5,1,1] => 27 [5,4,3] => 16 [5,4,2,1] => 24 [5,4,1,1,1] => 38 [5,3,3,1] => 25 [5,3,2,2] => 24 [5,3,2,1,1] => 38 [5,3,1,1,1,1] => 69 [5,2,2,2,1] => 42 [5,2,2,1,1,1] => 70 [5,2,1,1,1,1,1] => 132 [5,1,1,1,1,1,1,1] => 259 [4,4,4] => 28 [4,4,3,1] => 26 [4,4,2,2] => 36 [4,4,2,1,1] => 44 [4,4,1,1,1,1] => 72 [4,3,3,2] => 26 [4,3,3,1,1] => 40 [4,3,2,2,1] => 42 [4,3,2,1,1,1] => 70 [4,3,1,1,1,1,1] => 132 [4,2,2,2,2] => 64 [4,2,2,2,1,1] => 80 [4,2,2,1,1,1,1] => 136 [4,2,1,1,1,1,1,1] => 260 [4,1,1,1,1,1,1,1,1] => 514 [3,3,3,3] => 45 [3,3,3,2,1] => 46 [3,3,3,1,1,1] => 77 [3,3,2,2,2] => 44 [3,3,2,2,1,1] => 72 [3,3,2,1,1,1,1] => 134 [3,3,1,1,1,1,1,1] => 261 [3,2,2,2,2,1] => 80 [3,2,2,2,1,1,1] => 136 [3,2,2,1,1,1,1,1] => 260 [3,2,1,1,1,1,1,1,1] => 514 [3,1,1,1,1,1,1,1,1,1] => 1025 [2,2,2,2,2,2] => 126 [2,2,2,2,2,1,1] => 158 [2,2,2,2,1,1,1,1] => 270 [2,2,2,1,1,1,1,1,1] => 518 [2,2,1,1,1,1,1,1,1,1] => 1026 [2,1,1,1,1,1,1,1,1,1,1] => 2048 [1,1,1,1,1,1,1,1,1,1,1,1] => 4095 [13] => 13 [12,1] => 14 [11,2] => 14 [11,1,1] => 17 [10,3] => 14 [10,2,1] => 18 [10,1,1,1] => 24 [9,4] => 14 [9,3,1] => 19 [9,2,2] => 18 [9,2,1,1] => 24 [9,1,1,1,1] => 39 [8,5] => 14 [8,4,1] => 20 [8,3,2] => 18 [8,3,1,1] => 24 [8,2,2,1] => 28 [8,2,1,1,1] => 40 [8,1,1,1,1,1] => 70 [7,6] => 14 [7,5,1] => 17 [7,4,2] => 18 [7,4,1,1] => 24 [7,3,3] => 19 [7,3,2,1] => 24 [7,3,1,1,1] => 39 [7,2,2,2] => 28 [7,2,2,1,1] => 40 [7,2,1,1,1,1] => 70 [7,1,1,1,1,1,1] => 133 [6,6,1] => 22 [6,5,2] => 18 [6,5,1,1] => 24 [6,4,3] => 20 [6,4,2,1] => 28 [6,4,1,1,1] => 40 [6,3,3,1] => 32 [6,3,2,2] => 30 [6,3,2,1,1] => 42 [6,3,1,1,1,1] => 72 [6,2,2,2,1] => 50 [6,2,2,1,1,1] => 74 [6,2,1,1,1,1,1] => 134 [6,1,1,1,1,1,1,1] => 260 [5,5,3] => 21 [5,5,2,1] => 28 [5,5,1,1,1] => 43 [5,4,4] => 20 [5,4,3,1] => 24 [5,4,2,2] => 28 [5,4,2,1,1] => 40 [5,4,1,1,1,1] => 70 [5,3,3,2] => 26 [5,3,3,1,1] => 41 [5,3,2,2,1] => 40 [5,3,2,1,1,1] => 70 [5,3,1,1,1,1,1] => 133 [5,2,2,2,2] => 50 [5,2,2,2,1,1] => 74 [5,2,2,1,1,1,1] => 134 [5,2,1,1,1,1,1,1] => 260 [5,1,1,1,1,1,1,1,1] => 515 [4,4,4,1] => 36 [4,4,3,2] => 30 [4,4,3,1,1] => 42 [4,4,2,2,1] => 52 [4,4,2,1,1,1] => 76 [4,4,1,1,1,1,1] => 136 [4,3,3,3] => 32 [4,3,3,2,1] => 42 [4,3,3,1,1,1] => 72 [4,3,2,2,2] => 50 [4,3,2,2,1,1] => 74 [4,3,2,1,1,1,1] => 134 [4,3,1,1,1,1,1,1] => 260 [4,2,2,2,2,1] => 96 [4,2,2,2,1,1,1] => 144 [4,2,2,1,1,1,1,1] => 264 [4,2,1,1,1,1,1,1,1] => 516 [4,1,1,1,1,1,1,1,1,1] => 1026 [3,3,3,3,1] => 61 [3,3,3,2,2] => 48 [3,3,3,2,1,1] => 78 [3,3,3,1,1,1,1] => 141 [3,3,2,2,2,1] => 76 [3,3,2,2,1,1,1] => 136 [3,3,2,1,1,1,1,1] => 262 [3,3,1,1,1,1,1,1,1] => 517 [3,2,2,2,2,2] => 96 [3,2,2,2,2,1,1] => 144 [3,2,2,2,1,1,1,1] => 264 [3,2,2,1,1,1,1,1,1] => 516 [3,2,1,1,1,1,1,1,1,1] => 1026 [3,1,1,1,1,1,1,1,1,1,1] => 2049 [2,2,2,2,2,2,1] => 190 [2,2,2,2,2,1,1,1] => 286 [2,2,2,2,1,1,1,1,1] => 526 [2,2,2,1,1,1,1,1,1,1] => 1030 [2,2,1,1,1,1,1,1,1,1,1] => 2050 [2,1,1,1,1,1,1,1,1,1,1,1] => 4096 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 8191 [14] => 14 [13,1] => 15 [12,2] => 16 [12,1,1] => 18 [11,3] => 15 [11,2,1] => 18 [11,1,1,1] => 25 [10,4] => 16 [10,3,1] => 18 [10,2,2] => 22 [10,2,1,1] => 26 [10,1,1,1,1] => 40 [9,5] => 15 [9,4,1] => 18 [9,3,2] => 20 [9,3,1,1] => 27 [9,2,2,1] => 26 [9,2,1,1,1] => 40 [9,1,1,1,1,1] => 71 [8,6] => 16 [8,5,1] => 18 [8,4,2] => 24 [8,4,1,1] => 28 [8,3,3] => 20 [8,3,2,1] => 26 [8,3,1,1,1] => 40 [8,2,2,2] => 36 [8,2,2,1,1] => 44 [8,2,1,1,1,1] => 72 [8,1,1,1,1,1,1] => 134 [7,7] => 21 [7,6,1] => 18 [7,5,2] => 18 [7,5,1,1] => 25 [7,4,3] => 18 [7,4,2,1] => 26 [7,4,1,1,1] => 40 [7,3,3,1] => 27 [7,3,2,2] => 26 [7,3,2,1,1] => 40 [7,3,1,1,1,1] => 71 [7,2,2,2,1] => 44 [7,2,2,1,1,1] => 72 [7,2,1,1,1,1,1] => 134 [7,1,1,1,1,1,1,1] => 261 [6,6,2] => 26 [6,6,1,1] => 30 [6,5,3] => 20 [6,5,2,1] => 26 [6,5,1,1,1] => 40 [6,4,4] => 24 [6,4,3,1] => 28 [6,4,2,2] => 36 [6,4,2,1,1] => 44 [6,4,1,1,1,1] => 72 [6,3,3,2] => 34 [6,3,3,1,1] => 48 [6,3,2,2,1] => 46 [6,3,2,1,1,1] => 74 [6,3,1,1,1,1,1] => 136 [6,2,2,2,2] => 66 [6,2,2,2,1,1] => 82 [6,2,2,1,1,1,1] => 138 [6,2,1,1,1,1,1,1] => 262 [6,1,1,1,1,1,1,1,1] => 516 [5,5,4] => 22 [5,5,3,1] => 29 [5,5,2,2] => 30 [5,5,2,1,1] => 44 [5,5,1,1,1,1] => 75 [5,4,4,1] => 28 [5,4,3,2] => 26 [5,4,3,1,1] => 40 [5,4,2,2,1] => 44 [5,4,2,1,1,1] => 72 [5,4,1,1,1,1,1] => 134 [5,3,3,3] => 33 [5,3,3,2,1] => 42 [5,3,3,1,1,1] => 73 [5,3,2,2,2] => 44 [5,3,2,2,1,1] => 72 [5,3,2,1,1,1,1] => 134 [5,3,1,1,1,1,1,1] => 261 [5,2,2,2,2,1] => 82 [5,2,2,2,1,1,1] => 138 [5,2,2,1,1,1,1,1] => 262 [5,2,1,1,1,1,1,1,1] => 516 [5,1,1,1,1,1,1,1,1,1] => 1027 [4,4,4,2] => 44 [4,4,4,1,1] => 52 [4,4,3,3] => 30 [4,4,3,2,1] => 46 [4,4,3,1,1,1] => 74 [4,4,2,2,2] => 68 [4,4,2,2,1,1] => 84 [4,4,2,1,1,1,1] => 140 [4,4,1,1,1,1,1,1] => 264 [4,3,3,3,1] => 48 [4,3,3,2,2] => 46 [4,3,3,2,1,1] => 74 [4,3,3,1,1,1,1] => 136 [4,3,2,2,2,1] => 82 [4,3,2,2,1,1,1] => 138 [4,3,2,1,1,1,1,1] => 262 [4,3,1,1,1,1,1,1,1] => 516 [4,2,2,2,2,2] => 128 [4,2,2,2,2,1,1] => 160 [4,2,2,2,1,1,1,1] => 272 [4,2,2,1,1,1,1,1,1] => 520 [4,2,1,1,1,1,1,1,1,1] => 1028 [4,1,1,1,1,1,1,1,1,1,1] => 2050 [3,3,3,3,2] => 62 [3,3,3,3,1,1] => 93 [3,3,3,2,2,1] => 80 [3,3,3,2,1,1,1] => 142 [3,3,3,1,1,1,1,1] => 269 [3,3,2,2,2,2] => 84 [3,3,2,2,2,1,1] => 140 [3,3,2,2,1,1,1,1] => 264 [3,3,2,1,1,1,1,1,1] => 518 [3,3,1,1,1,1,1,1,1,1] => 1029 [3,2,2,2,2,2,1] => 160 [3,2,2,2,2,1,1,1] => 272 [3,2,2,2,1,1,1,1,1] => 520 [3,2,2,1,1,1,1,1,1,1] => 1028 [3,2,1,1,1,1,1,1,1,1,1] => 2050 [3,1,1,1,1,1,1,1,1,1,1,1] => 4097 [2,2,2,2,2,2,2] => 254 [2,2,2,2,2,2,1,1] => 318 [2,2,2,2,2,1,1,1,1] => 542 [2,2,2,2,1,1,1,1,1,1] => 1038 [2,2,2,1,1,1,1,1,1,1,1] => 2054 [2,2,1,1,1,1,1,1,1,1,1,1] => 4098 [2,1,1,1,1,1,1,1,1,1,1,1,1] => 8192 [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 16383 [15] => 15 [14,1] => 16 [13,2] => 16 [13,1,1] => 19 [12,3] => 18 [12,2,1] => 20 [12,1,1,1] => 26 [11,4] => 16 [11,3,1] => 19 [11,2,2] => 20 [11,2,1,1] => 26 [11,1,1,1,1] => 41 [10,5] => 20 [10,4,1] => 20 [10,3,2] => 20 [10,3,1,1] => 26 [10,2,2,1] => 30 [10,2,1,1,1] => 42 [10,1,1,1,1,1] => 72 [9,6] => 18 [9,5,1] => 19 [9,4,2] => 20 [9,4,1,1] => 26 [9,3,3] => 27 [9,3,2,1] => 28 [9,3,1,1,1] => 43 [9,2,2,2] => 30 [9,2,2,1,1] => 42 [9,2,1,1,1,1] => 72 [9,1,1,1,1,1,1] => 135 [8,7] => 16 [8,6,1] => 20 [8,5,2] => 20 [8,5,1,1] => 26 [8,4,3] => 22 [8,4,2,1] => 32 [8,4,1,1,1] => 44 [8,3,3,1] => 28 [8,3,2,2] => 30 [8,3,2,1,1] => 42 [8,3,1,1,1,1] => 72 [8,2,2,2,1] => 52 [8,2,2,1,1,1] => 76 [8,2,1,1,1,1,1] => 136 [8,1,1,1,1,1,1,1] => 262 [7,7,1] => 25 [7,6,2] => 20 [7,6,1,1] => 26 [7,5,3] => 19 [7,5,2,1] => 26 [7,5,1,1,1] => 41 [7,4,4] => 22 [7,4,3,1] => 26 [7,4,2,2] => 30 [7,4,2,1,1] => 42 [7,4,1,1,1,1] => 72 [7,3,3,2] => 28 [7,3,3,1,1] => 43 [7,3,2,2,1] => 42 [7,3,2,1,1,1] => 72 [7,3,1,1,1,1,1] => 135 [7,2,2,2,2] => 52 [7,2,2,2,1,1] => 76 [7,2,2,1,1,1,1] => 136 [7,2,1,1,1,1,1,1] => 262 [7,1,1,1,1,1,1,1,1] => 517 [6,6,3] => 30 [6,6,2,1] => 34 [6,6,1,1,1] => 46 [6,5,4] => 20 [6,5,3,1] => 28 [6,5,2,2] => 30 [6,5,2,1,1] => 42 [6,5,1,1,1,1] => 72 [6,4,4,1] => 32 [6,4,3,2] => 32 [6,4,3,1,1] => 44 [6,4,2,2,1] => 52 [6,4,2,1,1,1] => 76 [6,4,1,1,1,1,1] => 136 [6,3,3,3] => 48 [6,3,3,2,1] => 50 [6,3,3,1,1,1] => 80 [6,3,2,2,2] => 54 [6,3,2,2,1,1] => 78 [6,3,2,1,1,1,1] => 138 [6,3,1,1,1,1,1,1] => 264 [6,2,2,2,2,1] => 98 [6,2,2,2,1,1,1] => 146 [6,2,2,1,1,1,1,1] => 266 [6,2,1,1,1,1,1,1,1] => 518 [6,1,1,1,1,1,1,1,1,1] => 1028 [5,5,5] => 35 [5,5,4,1] => 30 [5,5,3,2] => 30 [5,5,3,1,1] => 45 [5,5,2,2,1] => 46 [5,5,2,1,1,1] => 76 [5,5,1,1,1,1,1] => 139 [5,4,4,2] => 32 [5,4,4,1,1] => 44 [5,4,3,3] => 28 [5,4,3,2,1] => 42 [5,4,3,1,1,1] => 72 [5,4,2,2,2] => 52 [5,4,2,2,1,1] => 76 [5,4,2,1,1,1,1] => 136 [5,4,1,1,1,1,1,1] => 262 [5,3,3,3,1] => 49 [5,3,3,2,2] => 44 [5,3,3,2,1,1] => 74 [5,3,3,1,1,1,1] => 137 [5,3,2,2,2,1] => 76 [5,3,2,2,1,1,1] => 136 [5,3,2,1,1,1,1,1] => 262 [5,3,1,1,1,1,1,1,1] => 517 [5,2,2,2,2,2] => 98 [5,2,2,2,2,1,1] => 146 [5,2,2,2,1,1,1,1] => 266 [5,2,2,1,1,1,1,1,1] => 518 [5,2,1,1,1,1,1,1,1,1] => 1028 [5,1,1,1,1,1,1,1,1,1,1] => 2051 [4,4,4,3] => 38 [4,4,4,2,1] => 60 [4,4,4,1,1,1] => 84 [4,4,3,3,1] => 46 [4,4,3,2,2] => 54 [4,4,3,2,1,1] => 78 [4,4,3,1,1,1,1] => 138 [4,4,2,2,2,1] => 100 [4,4,2,2,1,1,1] => 148 [4,4,2,1,1,1,1,1] => 268 [4,4,1,1,1,1,1,1,1] => 520 [4,3,3,3,2] => 50 [4,3,3,3,1,1] => 80 [4,3,3,2,2,1] => 78 [4,3,3,2,1,1,1] => 138 [4,3,3,1,1,1,1,1] => 264 [4,3,2,2,2,2] => 98 [4,3,2,2,2,1,1] => 146 [4,3,2,2,1,1,1,1] => 266 [4,3,2,1,1,1,1,1,1] => 518 [4,3,1,1,1,1,1,1,1,1] => 1028 [4,2,2,2,2,2,1] => 192 [4,2,2,2,2,1,1,1] => 288 [4,2,2,2,1,1,1,1,1] => 528 [4,2,2,1,1,1,1,1,1,1] => 1032 [4,2,1,1,1,1,1,1,1,1,1] => 2052 [4,1,1,1,1,1,1,1,1,1,1,1] => 4098 [3,3,3,3,3] => 93 [3,3,3,3,2,1] => 94 [3,3,3,3,1,1,1] => 157 [3,3,3,2,2,2] => 84 [3,3,3,2,2,1,1] => 144 [3,3,3,2,1,1,1,1] => 270 [3,3,3,1,1,1,1,1,1] => 525 [3,3,2,2,2,2,1] => 148 [3,3,2,2,2,1,1,1] => 268 [3,3,2,2,1,1,1,1,1] => 520 [3,3,2,1,1,1,1,1,1,1] => 1030 [3,3,1,1,1,1,1,1,1,1,1] => 2053 [3,2,2,2,2,2,2] => 192 [3,2,2,2,2,2,1,1] => 288 [3,2,2,2,2,1,1,1,1] => 528 [3,2,2,2,1,1,1,1,1,1] => 1032 [3,2,2,1,1,1,1,1,1,1,1] => 2052 [3,2,1,1,1,1,1,1,1,1,1,1] => 4098 [3,1,1,1,1,1,1,1,1,1,1,1,1] => 8193 [2,2,2,2,2,2,2,1] => 382 [2,2,2,2,2,2,1,1,1] => 574 [2,2,2,2,2,1,1,1,1,1] => 1054 [2,2,2,2,1,1,1,1,1,1,1] => 2062 [2,2,2,1,1,1,1,1,1,1,1,1] => 4102 [2,2,1,1,1,1,1,1,1,1,1,1,1] => 8194 [2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 16384 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 32767 [16] => 16 [15,1] => 17 [14,2] => 18 [14,1,1] => 20 [13,3] => 17 [13,2,1] => 20 [13,1,1,1] => 27 [12,4] => 20 [12,3,1] => 22 [12,2,2] => 24 [12,2,1,1] => 28 [12,1,1,1,1] => 42 [11,5] => 17 [11,4,1] => 20 [11,3,2] => 20 [11,3,1,1] => 27 [11,2,2,1] => 28 [11,2,1,1,1] => 42 [11,1,1,1,1,1] => 73 [10,6] => 18 [10,5,1] => 24 [10,4,2] => 24 [10,4,1,1] => 28 [10,3,3] => 22 [10,3,2,1] => 28 [10,3,1,1,1] => 42 [10,2,2,2] => 38 [10,2,2,1,1] => 46 [10,2,1,1,1,1] => 74 [10,1,1,1,1,1,1] => 136 [9,7] => 17 [9,6,1] => 22 [9,5,2] => 20 [9,5,1,1] => 27 [9,4,3] => 22 [9,4,2,1] => 28 [9,4,1,1,1] => 42 [9,3,3,1] => 35 [9,3,2,2] => 30 [9,3,2,1,1] => 44 [9,3,1,1,1,1] => 75 [9,2,2,2,1] => 46 [9,2,2,1,1,1] => 74 [9,2,1,1,1,1,1] => 136 [9,1,1,1,1,1,1,1] => 263 [8,8] => 24 [8,7,1] => 20 [8,6,2] => 24 [8,6,1,1] => 28 [8,5,3] => 20 [8,5,2,1] => 28 [8,5,1,1,1] => 42 [8,4,4] => 32 [8,4,3,1] => 30 [8,4,2,2] => 40 [8,4,2,1,1] => 48 [8,4,1,1,1,1] => 76 [8,3,3,2] => 30 [8,3,3,1,1] => 44 [8,3,2,2,1] => 46 [8,3,2,1,1,1] => 74 [8,3,1,1,1,1,1] => 136 [8,2,2,2,2] => 68 [8,2,2,2,1,1] => 84 [8,2,2,1,1,1,1] => 140 [8,2,1,1,1,1,1,1] => 264 [8,1,1,1,1,1,1,1,1] => 518 [7,7,2] => 26 [7,7,1,1] => 33 [7,6,3] => 22 [7,6,2,1] => 28 [7,6,1,1,1] => 42 [7,5,4] => 20 [7,5,3,1] => 27 [7,5,2,2] => 28 [7,5,2,1,1] => 42 [7,5,1,1,1,1] => 73 [7,4,4,1] => 30 [7,4,3,2] => 28 [7,4,3,1,1] => 42 [7,4,2,2,1] => 46 [7,4,2,1,1,1] => 74 [7,4,1,1,1,1,1] => 136 [7,3,3,3] => 35 [7,3,3,2,1] => 44 [7,3,3,1,1,1] => 75 [7,3,2,2,2] => 46 [7,3,2,2,1,1] => 74 [7,3,2,1,1,1,1] => 136 [7,3,1,1,1,1,1,1] => 263 [7,2,2,2,2,1] => 84 [7,2,2,2,1,1,1] => 140 [7,2,2,1,1,1,1,1] => 264 [7,2,1,1,1,1,1,1,1] => 518 [7,1,1,1,1,1,1,1,1,1] => 1029 [6,6,4] => 28 [6,6,3,1] => 38 [6,6,2,2] => 42 [6,6,2,1,1] => 50 [6,6,1,1,1,1] => 78 [6,5,5] => 24 [6,5,4,1] => 28 [6,5,3,2] => 30 [6,5,3,1,1] => 44 [6,5,2,2,1] => 46 [6,5,2,1,1,1] => 74 [6,5,1,1,1,1,1] => 136 [6,4,4,2] => 40 [6,4,4,1,1] => 48 [6,4,3,3] => 36 [6,4,3,2,1] => 48 [6,4,3,1,1,1] => 76 [6,4,2,2,2] => 68 [6,4,2,2,1,1] => 84 [6,4,2,1,1,1,1] => 140 [6,4,1,1,1,1,1,1] => 264 [6,3,3,3,1] => 64 [6,3,3,2,2] => 54 [6,3,3,2,1,1] => 82 [6,3,3,1,1,1,1] => 144 [6,3,2,2,2,1] => 86 [6,3,2,2,1,1,1] => 142 [6,3,2,1,1,1,1,1] => 266 [6,3,1,1,1,1,1,1,1] => 520 [6,2,2,2,2,2] => 130 [6,2,2,2,2,1,1] => 162 [6,2,2,2,1,1,1,1] => 274 [6,2,2,1,1,1,1,1,1] => 522 [6,2,1,1,1,1,1,1,1,1] => 1030 [6,1,1,1,1,1,1,1,1,1,1] => 2052 [5,5,5,1] => 43 [5,5,4,2] => 32 [5,5,4,1,1] => 46 [5,5,3,3] => 33 [5,5,3,2,1] => 46 [5,5,3,1,1,1] => 77 [5,5,2,2,2] => 50 [5,5,2,2,1,1] => 78 [5,5,2,1,1,1,1] => 140 [5,5,1,1,1,1,1,1] => 267 [5,4,4,3] => 30 [5,4,4,2,1] => 48 [5,4,4,1,1,1] => 76 [5,4,3,3,1] => 44 [5,4,3,2,2] => 46 [5,4,3,2,1,1] => 74 [5,4,3,1,1,1,1] => 136 [5,4,2,2,2,1] => 84 [5,4,2,2,1,1,1] => 140 [5,4,2,1,1,1,1,1] => 264 [5,4,1,1,1,1,1,1,1] => 518 [5,3,3,3,2] => 50 [5,3,3,3,1,1] => 81 [5,3,3,2,2,1] => 76 [5,3,3,2,1,1,1] => 138 [5,3,3,1,1,1,1,1] => 265 [5,3,2,2,2,2] => 84 [5,3,2,2,2,1,1] => 140 [5,3,2,2,1,1,1,1] => 264 [5,3,2,1,1,1,1,1,1] => 518 [5,3,1,1,1,1,1,1,1,1] => 1029 [5,2,2,2,2,2,1] => 162 [5,2,2,2,2,1,1,1] => 274 [5,2,2,2,1,1,1,1,1] => 522 [5,2,2,1,1,1,1,1,1,1] => 1030 [5,2,1,1,1,1,1,1,1,1,1] => 2052 [5,1,1,1,1,1,1,1,1,1,1,1] => 4099 [4,4,4,4] => 60 [4,4,4,3,1] => 54 [4,4,4,2,2] => 76 [4,4,4,2,1,1] => 92 [4,4,4,1,1,1,1] => 148 [4,4,3,3,2] => 50 [4,4,3,3,1,1] => 78 [4,4,3,2,2,1] => 86 [4,4,3,2,1,1,1] => 142 [4,4,3,1,1,1,1,1] => 266 [4,4,2,2,2,2] => 132 [4,4,2,2,2,1,1] => 164 [4,4,2,2,1,1,1,1] => 276 [4,4,2,1,1,1,1,1,1] => 524 [4,4,1,1,1,1,1,1,1,1] => 1032 [4,3,3,3,3] => 64 [4,3,3,3,2,1] => 82 [4,3,3,3,1,1,1] => 144 [4,3,3,2,2,2] => 86 [4,3,3,2,2,1,1] => 142 [4,3,3,2,1,1,1,1] => 266 [4,3,3,1,1,1,1,1,1] => 520 [4,3,2,2,2,2,1] => 162 [4,3,2,2,2,1,1,1] => 274 [4,3,2,2,1,1,1,1,1] => 522 [4,3,2,1,1,1,1,1,1,1] => 1030 [4,3,1,1,1,1,1,1,1,1,1] => 2052 [4,2,2,2,2,2,2] => 256 [4,2,2,2,2,2,1,1] => 320 [4,2,2,2,2,1,1,1,1] => 544 [4,2,2,2,1,1,1,1,1,1] => 1040 [4,2,2,1,1,1,1,1,1,1,1] => 2056 [4,2,1,1,1,1,1,1,1,1,1,1] => 4100 [4,1,1,1,1,1,1,1,1,1,1,1,1] => 8194 [3,3,3,3,3,1] => 125 [3,3,3,3,2,2] => 96 [3,3,3,3,2,1,1] => 158 [3,3,3,3,1,1,1,1] => 285 [3,3,3,2,2,2,1] => 148 [3,3,3,2,2,1,1,1] => 272 [3,3,3,2,1,1,1,1,1] => 526 [3,3,3,1,1,1,1,1,1,1] => 1037 [3,3,2,2,2,2,2] => 164 [3,3,2,2,2,2,1,1] => 276 [3,3,2,2,2,1,1,1,1] => 524 [3,3,2,2,1,1,1,1,1,1] => 1032 [3,3,2,1,1,1,1,1,1,1,1] => 2054 [3,3,1,1,1,1,1,1,1,1,1,1] => 4101 [3,2,2,2,2,2,2,1] => 320 [3,2,2,2,2,2,1,1,1] => 544 [3,2,2,2,2,1,1,1,1,1] => 1040 [3,2,2,2,1,1,1,1,1,1,1] => 2056 [3,2,2,1,1,1,1,1,1,1,1,1] => 4100 [3,2,1,1,1,1,1,1,1,1,1,1,1] => 8194 [3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 16385 [2,2,2,2,2,2,2,2] => 510 [2,2,2,2,2,2,2,1,1] => 638 [2,2,2,2,2,2,1,1,1,1] => 1086 [2,2,2,2,2,1,1,1,1,1,1] => 2078 [2,2,2,2,1,1,1,1,1,1,1,1] => 4110 [2,2,2,1,1,1,1,1,1,1,1,1,1] => 8198 [2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 16386 [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 32768 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 65535 [17] => 17 [16,1] => 18 [15,2] => 18 [15,1,1] => 21 [14,3] => 18 [14,2,1] => 22 [14,1,1,1] => 28 [13,4] => 18 [13,3,1] => 21 [13,2,2] => 22 [13,2,1,1] => 28 [13,1,1,1,1] => 43 [12,5] => 18 [12,4,1] => 24 [12,3,2] => 24 [12,3,1,1] => 30 [12,2,2,1] => 32 [12,2,1,1,1] => 44 [12,1,1,1,1,1] => 74 [11,6] => 18 [11,5,1] => 21 [11,4,2] => 22 [11,4,1,1] => 28 [11,3,3] => 23 [11,3,2,1] => 28 [11,3,1,1,1] => 43 [11,2,2,2] => 32 [11,2,2,1,1] => 44 [11,2,1,1,1,1] => 74 [11,1,1,1,1,1,1] => 137 [10,7] => 18 [10,6,1] => 22 [10,5,2] => 26 [10,5,1,1] => 32 [10,4,3] => 22 [10,4,2,1] => 32 [10,4,1,1,1] => 44 [10,3,3,1] => 30 [10,3,2,2] => 32 [10,3,2,1,1] => 44 [10,3,1,1,1,1] => 74 [10,2,2,2,1] => 54 [10,2,2,1,1,1] => 78 [10,2,1,1,1,1,1] => 138 [10,1,1,1,1,1,1,1] => 264 [9,8] => 18 [9,7,1] => 21 [9,6,2] => 24 [9,6,1,1] => 30 [9,5,3] => 23 [9,5,2,1] => 28 [9,5,1,1,1] => 43 [9,4,4] => 24 [9,4,3,1] => 30 [9,4,2,2] => 32 [9,4,2,1,1] => 44 [9,4,1,1,1,1] => 74 [9,3,3,2] => 36 [9,3,3,1,1] => 51 [9,3,2,2,1] => 46 [9,3,2,1,1,1] => 76 [9,3,1,1,1,1,1] => 139 [9,2,2,2,2] => 54 [9,2,2,2,1,1] => 78 [9,2,2,1,1,1,1] => 138 [9,2,1,1,1,1,1,1] => 264 [9,1,1,1,1,1,1,1,1] => 519 [8,8,1] => 28 [8,7,2] => 22 [8,7,1,1] => 28 [8,6,3] => 24 [8,6,2,1] => 32 [8,6,1,1,1] => 44 [8,5,4] => 24 [8,5,3,1] => 28 [8,5,2,2] => 32 [8,5,2,1,1] => 44 [8,5,1,1,1,1] => 74 [8,4,4,1] => 40 [8,4,3,2] => 34 [8,4,3,1,1] => 46 [8,4,2,2,1] => 56 [8,4,2,1,1,1] => 80 [8,4,1,1,1,1,1] => 140 [8,3,3,3] => 36 [8,3,3,2,1] => 46 [8,3,3,1,1,1] => 76 [8,3,2,2,2] => 54 [8,3,2,2,1,1] => 78 [8,3,2,1,1,1,1] => 138 [8,3,1,1,1,1,1,1] => 264 [8,2,2,2,2,1] => 100 [8,2,2,2,1,1,1] => 148 [8,2,2,1,1,1,1,1] => 268 [8,2,1,1,1,1,1,1,1] => 520 [8,1,1,1,1,1,1,1,1,1] => 1030 [7,7,3] => 27 [7,7,2,1] => 34 [7,7,1,1,1] => 49 [7,6,4] => 22 [7,6,3,1] => 30 [7,6,2,2] => 32 [7,6,2,1,1] => 44 [7,6,1,1,1,1] => 74 [7,5,5] => 25 [7,5,4,1] => 28 [7,5,3,2] => 28 [7,5,3,1,1] => 43 [7,5,2,2,1] => 44 [7,5,2,1,1,1] => 74 [7,5,1,1,1,1,1] => 137 [7,4,4,2] => 34 [7,4,4,1,1] => 46 [7,4,3,3] => 30 [7,4,3,2,1] => 44 [7,4,3,1,1,1] => 74 [7,4,2,2,2] => 54 [7,4,2,2,1,1] => 78 [7,4,2,1,1,1,1] => 138 [7,4,1,1,1,1,1,1] => 264 [7,3,3,3,1] => 51 [7,3,3,2,2] => 46 [7,3,3,2,1,1] => 76 [7,3,3,1,1,1,1] => 139 [7,3,2,2,2,1] => 78 [7,3,2,2,1,1,1] => 138 [7,3,2,1,1,1,1,1] => 264 [7,3,1,1,1,1,1,1,1] => 519 [7,2,2,2,2,2] => 100 [7,2,2,2,2,1,1] => 148 [7,2,2,2,1,1,1,1] => 268 [7,2,2,1,1,1,1,1,1] => 520 [7,2,1,1,1,1,1,1,1,1] => 1030 [7,1,1,1,1,1,1,1,1,1,1] => 2053 [6,6,5] => 26 [6,6,4,1] => 36 [6,6,3,2] => 42 [6,6,3,1,1] => 54 [6,6,2,2,1] => 58 [6,6,2,1,1,1] => 82 [6,6,1,1,1,1,1] => 142 [6,5,5,1] => 32 [6,5,4,2] => 32 [6,5,4,1,1] => 44 [6,5,3,3] => 36 [6,5,3,2,1] => 46 [6,5,3,1,1,1] => 76 [6,5,2,2,2] => 54 [6,5,2,2,1,1] => 78 [6,5,2,1,1,1,1] => 138 [6,5,1,1,1,1,1,1] => 264 [6,4,4,3] => 36 [6,4,4,2,1] => 56 [6,4,4,1,1,1] => 80 [6,4,3,3,1] => 52 [6,4,3,2,2] => 56 [6,4,3,2,1,1] => 80 [6,4,3,1,1,1,1] => 140 [6,4,2,2,2,1] => 100 [6,4,2,2,1,1,1] => 148 [6,4,2,1,1,1,1,1] => 268 [6,4,1,1,1,1,1,1,1] => 520 [6,3,3,3,2] => 66 [6,3,3,3,1,1] => 96 [6,3,3,2,2,1] => 86 [6,3,3,2,1,1,1] => 146 [6,3,3,1,1,1,1,1] => 272 [6,3,2,2,2,2] => 102 [6,3,2,2,2,1,1] => 150 [6,3,2,2,1,1,1,1] => 270 [6,3,2,1,1,1,1,1,1] => 522 [6,3,1,1,1,1,1,1,1,1] => 1032 [6,2,2,2,2,2,1] => 194 [6,2,2,2,2,1,1,1] => 290 [6,2,2,2,1,1,1,1,1] => 530 [6,2,2,1,1,1,1,1,1,1] => 1034 [6,2,1,1,1,1,1,1,1,1,1] => 2054 [6,1,1,1,1,1,1,1,1,1,1,1] => 4100 [5,5,5,2] => 44 [5,5,5,1,1] => 59 [5,5,4,3] => 32 [5,5,4,2,1] => 48 [5,5,4,1,1,1] => 78 [5,5,3,3,1] => 49 [5,5,3,2,2] => 48 [5,5,3,2,1,1] => 78 [5,5,3,1,1,1,1] => 141 [5,5,2,2,2,1] => 82 [5,5,2,2,1,1,1] => 142 [5,5,2,1,1,1,1,1] => 268 [5,5,1,1,1,1,1,1,1] => 523 [5,4,4,4] => 40 [5,4,4,3,1] => 46 [5,4,4,2,2] => 56 [5,4,4,2,1,1] => 80 [5,4,4,1,1,1,1] => 140 [5,4,3,3,2] => 46 [5,4,3,3,1,1] => 76 [5,4,3,2,2,1] => 78 [5,4,3,2,1,1,1] => 138 [5,4,3,1,1,1,1,1] => 264 [5,4,2,2,2,2] => 100 [5,4,2,2,2,1,1] => 148 [5,4,2,2,1,1,1,1] => 268 [5,4,2,1,1,1,1,1,1] => 520 [5,4,1,1,1,1,1,1,1,1] => 1030 [5,3,3,3,3] => 65 [5,3,3,3,2,1] => 82 [5,3,3,3,1,1,1] => 145 [5,3,3,2,2,2] => 80 [5,3,3,2,2,1,1] => 140 [5,3,3,2,1,1,1,1] => 266 [5,3,3,1,1,1,1,1,1] => 521 [5,3,2,2,2,2,1] => 148 [5,3,2,2,2,1,1,1] => 268 [5,3,2,2,1,1,1,1,1] => 520 [5,3,2,1,1,1,1,1,1,1] => 1030 [5,3,1,1,1,1,1,1,1,1,1] => 2053 [5,2,2,2,2,2,2] => 194 [5,2,2,2,2,2,1,1] => 290 [5,2,2,2,2,1,1,1,1] => 530 [5,2,2,2,1,1,1,1,1,1] => 1034 [5,2,2,1,1,1,1,1,1,1,1] => 2054 [5,2,1,1,1,1,1,1,1,1,1,1] => 4100 [5,1,1,1,1,1,1,1,1,1,1,1,1] => 8195 [4,4,4,4,1] => 76 [4,4,4,3,2] => 62 [4,4,4,3,1,1] => 86 [4,4,4,2,2,1] => 108 [4,4,4,2,1,1,1] => 156 [4,4,4,1,1,1,1,1] => 276 [4,4,3,3,3] => 54 [4,4,3,3,2,1] => 82 [4,4,3,3,1,1,1] => 142 [4,4,3,2,2,2] => 102 [4,4,3,2,2,1,1] => 150 [4,4,3,2,1,1,1,1] => 270 [4,4,3,1,1,1,1,1,1] => 522 [4,4,2,2,2,2,1] => 196 [4,4,2,2,2,1,1,1] => 292 [4,4,2,2,1,1,1,1,1] => 532 [4,4,2,1,1,1,1,1,1,1] => 1036 [4,4,1,1,1,1,1,1,1,1,1] => 2056 [4,3,3,3,3,1] => 96 [4,3,3,3,2,2] => 86 [4,3,3,3,2,1,1] => 146 [4,3,3,3,1,1,1,1] => 272 [4,3,3,2,2,2,1] => 150 [4,3,3,2,2,1,1,1] => 270 [4,3,3,2,1,1,1,1,1] => 522 [4,3,3,1,1,1,1,1,1,1] => 1032 [4,3,2,2,2,2,2] => 194 [4,3,2,2,2,2,1,1] => 290 [4,3,2,2,2,1,1,1,1] => 530 [4,3,2,2,1,1,1,1,1,1] => 1034 [4,3,2,1,1,1,1,1,1,1,1] => 2054 [4,3,1,1,1,1,1,1,1,1,1,1] => 4100 [4,2,2,2,2,2,2,1] => 384 [4,2,2,2,2,2,1,1,1] => 576 [4,2,2,2,2,1,1,1,1,1] => 1056 [4,2,2,2,1,1,1,1,1,1,1] => 2064 [4,2,2,1,1,1,1,1,1,1,1,1] => 4104 [4,2,1,1,1,1,1,1,1,1,1,1,1] => 8196 [4,1,1,1,1,1,1,1,1,1,1,1,1,1] => 16386 [3,3,3,3,3,2] => 126 [3,3,3,3,3,1,1] => 189 [3,3,3,3,2,2,1] => 160 [3,3,3,3,2,1,1,1] => 286 [3,3,3,3,1,1,1,1,1] => 541 [3,3,3,2,2,2,2] => 156 [3,3,3,2,2,2,1,1] => 276 [3,3,3,2,2,1,1,1,1] => 528 [3,3,3,2,1,1,1,1,1,1] => 1038 [3,3,3,1,1,1,1,1,1,1,1] => 2061 [3,3,2,2,2,2,2,1] => 292 [3,3,2,2,2,2,1,1,1] => 532 [3,3,2,2,2,1,1,1,1,1] => 1036 [3,3,2,2,1,1,1,1,1,1,1] => 2056 [3,3,2,1,1,1,1,1,1,1,1,1] => 4102 [3,3,1,1,1,1,1,1,1,1,1,1,1] => 8197 [3,2,2,2,2,2,2,2] => 384 [3,2,2,2,2,2,2,1,1] => 576 [3,2,2,2,2,2,1,1,1,1] => 1056 [3,2,2,2,2,1,1,1,1,1,1] => 2064 [3,2,2,2,1,1,1,1,1,1,1,1] => 4104 [3,2,2,1,1,1,1,1,1,1,1,1,1] => 8196 ----------------------------------------------------------------------------- Created: Oct 24, 2025 at 14:03 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Oct 24, 2025 at 14:03 by Martin Rubey