六次等幂和

northwolves · 发表于 4 天前

n=123;IntegerPartitions[n^6,{9},Range[n-1]^6]^(1/6)
{{112,106,62,46,40,22,8,5,4},{110,101,86,74,62,34,34,32,14}}

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mathe · 发表于 3 天前

n=88,k最小150。

northwolves · 发表于 3 天前

$150^6= 109^6+104^6+100^6+96^6+94^6+92^6+89^6+88^6+86^6+85^6+84^6+83^6+82^6+80^6+79^6+78^6+76^6+74^6+73^6+72^6+71^6+70^6+69^6+68^6+67^6+66^6+65^6+64^6+62^6+61^6+60^6+59^6+58^6+56^6+55^6+54^6+53^6+52^6+51^6+50^6+49^6+48^6+47^6+46^6+45^6+44^6+43^6+42^6+41^6+40^6+39^6+38^6+37^6+36^6+35^6+33^6+32^6+31^6+30^6+29^6+28^6+27^6+26^6+25^6+24^6+23^6+22^6+21^6+20^6+19^6+18^6+17^6+16^6+15^6+14^6+13^6+12^6+11^6+10^6+9^6+8^6+7^6+6^6+5^6+4^6+3^6+2^6+1^6$

mathe · 发表于前天 07:44

n=102大概率是183，已经验证173以下都不是。
m=183, r=['154', '118', '113', '112', '106', '105', '103', '101', '100', '98', '97', '96', '95', '94', '91', '90', '89', '88', '87', '86', '85', '84', '83', '82', '81', '80', '79', '78', '77', '76', '75', '74', '73', '72', '70', '69', '68', '67', '66', '65', '64', '63', '62', '61', '60', '59', '58', '57', '56', '55', '54', '53', '52', '50', '49', '48', '47', '46', '45', '44', '43', '42', '41', '40', '39', '38', '37', '36', '35', '34', '33', '32', '31', '30', '29', '28', '26', '25', '24', '23', '22', '21', '20', '19', '18', '17', '16', '15', '14', '13', '12', '11', '10', '9', '8', '7', '6', '5', '4', '3', '2', '1']

mathe · 发表于前天 16:57

m=176 126 112 111 110 109 108 107 105 102 100 99 98 96 95 93 92 91 90 89 88 87 86 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 46 45 44 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1

mathe · 发表于前天 18:43

最终结果,m=175
=== RESULT ===
n=102 m=175
Solution: 123 114 111 108 106 105 104 103 102 101 100 99 97 96 95 93 92 90 89 88 87 86 85 84 83 82 81 80 79 78 76 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 58 57 56 55 54 53 52 51 50 48 47 46 45 44 43 42 41 40 39 38 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

mathe · 发表于昨天 07:06

现在汇总一下6次方的进度,100一下wolves已经提交到OEIS,这里就不给了
101    171       N/A
102    175
103    175       N/A
104    175       N/A
105    175       N/A
106    178       N/A
107    179       N/A
108    181       N/A
109    183       N/A
110    185       N/A
111    188       N/A
112    189       N/A
113    190       N/A
114    191       N/A
115    195       N/A
116    197       N/A
117    198       N/A
118    202       N/A
119    205
120    205       N/A
121    206       N/A
122    208       N/A
123    208       N/A
124    211       N/A
125    212       N/A
126    217
127    217       N/A
128    221
129    222       N/A
130    223       N/A
131    224       N/A
132    227       N/A
133    228       N/A
134    232
135    234-237
136    234       N/A
137    237
138    238       N/A
139    239       N/A
140    241       N/A
141    244       N/A
142    246-247
143    249-
144    251
145    251       N/A
146    251       N/A
147    254       N/A
148    255       N/A
149    257       N/A
150    261       N/A
其中标记了N/A的结果相对比较容易构造，就不需要额外记录了。
n=119, m=205
FOUND 205 136 134 129 125 124 123 122 121 120 118 116 115 114 113 110 109 107 106 104 103 102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 76 75 74 73 72 70 69 68 67 66 65 64 63 62 61 60 59 58 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

n=128,m=221
FOUND 221 144 141 137 135 133 132 130 128 127 126 124 123 122 121 120 118 117 116 115 114 113 111 109 108 107 106 105 104 103 102 101 100 99 98 97 96 95 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

n=134 m=232
Solution: 156 154 152 147 140 136 134 133 132 131 130 128 125 124 122 120 119 118 117 115 114 113 112 111 110 109 108 107 106 105 104 103 102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

n=135,m<=237
*** Found solution: m=237, r=['176', '146', '145', '144', '141', '135', '134', '133', '132', '131', '130', '129', '128', '127', '126', '124', '123', '122', '121', '120', '118', '117', '116', '115', '114', '113', '112', '111', '110', '109', '108', '107', '106', '104', '103', '102', '101', '100', '99', '98', '97', '96', '95', '94', '93', '92', '91', '90', '89', '88', '87', '86', '85', '84', '83', '82', '81', '80', '79', '78', '77', '76', '75', '74', '73', '72', '71', '70', '69', '68', '67', '66', '65', '64', '62', '61', '60', '59', '58', '57', '56', '55', '54', '53', '52', '51', '50', '49', '48', '47', '46', '45', '44', '43', '42', '41', '40', '39', '38', '37', '36', '34', '33', '32', '31', '30', '29', '28', '27', '26', '25', '24', '23', '22', '21', '20', '19', '18', '17', '16', '15', '14', '13', '12', '11', '10', '9', '8', '7', '6', '5', '4', '3', '2', '1']

n=137 m=237
Solution: 151 147 144 143 141 139 138 137 135 134 133 131 130 129 128 127 125 124 123 122 121 120 119 118 117 116 115 114 113 112 111 110 109 108 107 106 105 104 103 102 101 100 99 97 96 95 94 93 92 91 90 89 88 87 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 34 33 32 31 30 29 28 27 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

n=142, m<=247
*** Found solution: m=247, r=['165', '161', '155', '153', '148', '147', '144', '141', '140', '138', '136', '135', '133', '132', '130', '129', '128', '127', '126', '125', '124', '123', '121', '120', '119', '118', '117', '116', '115', '114', '113', '112', '111', '110', '109', '108', '107', '106', '105', '104', '103', '102', '101', '99', '98', '97', '96', '95', '94', '93', '92', '91', '90', '89', '88', '87', '86', '85', '84', '83', '82', '81', '80', '79', '78', '77', '76', '75', '74', '73', '72', '71', '70', '69', '68', '67', '66', '65', '64', '63', '62', '61', '60', '59', '58', '57', '56', '55', '54', '53', '52', '51', '50', '49', '48', '47', '46', '45', '44', '43', '42', '41', '40', '39', '38', '37', '36', '35', '34', '33', '32', '31', '30', '29', '28', '27', '26', '25', '24', '23', '22', '21', '20', '19', '18', '17', '16', '15', '14', '13', '12', '11', '10', '9', '8', '7', '6', '5', '4', '3', '2', '1']

n=144 m=251
Solution: 170 155 150 149 148 146 145 144 143 142 139 138 137 136 135 134 133 131 130 129 128 127 125 124 123 122 121 119 118 117 116 115 114 113 112 111 110 109 108 107 106 105 104 103 102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

mathe · 发表于昨天 11:37

n=135 m=234
FOUND 234 151 144 142 141 139 138 137 136 134 132 131 130 129 128 127 126 125 124 123 122 121 120 118 116 115 114 113 112 110 109 108 107 106 105 104 103 102 101 100 99 98 97 96 95 94 93 92 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

n=142, m=247, 所以现在对于k=6，n=142都有结果了。但是n=143看上去还有差距

mathe · 发表于昨天 20:11

n=143, m=250
FOUND 250 161 154 151 148 147 145 144 143 142 141 140 139 138 137 136 134 133 131 130 129 128 127 126 125 124 123 122 121 120 119 118 117 116 115 113 112 110 109 108 107 106 105 104 103 102 101 100 99 98 97 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

所以到n=150都确定了
101    171       171
102    175
103    175       N/A
104    175       N/A
105    175       N/A
106    178       N/A
107    179       N/A
108    181       N/A
109    183       N/A
110    185       N/A
111    188       N/A
112    189       N/A
113    190       N/A
114    191       N/A
115    195       N/A
116    197       N/A
117    198       N/A
118    202       N/A
119    205
120    205       N/A
121    206       N/A
122    208       N/A
123    208       N/A
124    211       N/A
125    212       N/A
126    217
127    217       N/A
128    221
129    222       N/A
130    223       N/A
131    224       N/A
132    227       N/A
133    228       N/A
134    232
135    234
136    234       N/A
137    237
138    238       N/A
139    239       N/A
140    241       N/A
141    244       N/A
142    247
143    250
144    251
145    251       N/A
146    251       N/A
147    254       N/A
148    255       N/A
149    257       N/A
150    261       N/A

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