タクシー数の一覧(list of Taxi-cab numbers)

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 タクシー数の定義

「タクシー数」は「ハーディ-ラマヌジャン数」とも呼ばれ、一般に「2つの正の立方数の和として表すことができる最小の整数」(A011541)として定義されますが、ここでは「最小の」という部分を取り払って「少なくとも2通りの正の立方数の和で表せる整数」(A001235)をタクシー数(Taxi-cab numbers)と呼ぶことにします。

※Wikipediaでは前者の定義を採用していますが、オンライン整数列辞典OEISでは “Taxi-cab numbers” を上記の2通りの解釈で別々の数列として登録されています。

 

「タクシー数」一覧

以下、$N=a^3+b^3=c^3+d^3$ を満たすような整数組$(a,b,c,d)$と、そのような組が存在する$N$のリストを掲載します。$a$~$d$が$10^3$以下の範囲で探索した結果を示しています。$N$の列はオンライン整数列辞典のA001235にほぼ一致しているはずです。

また、このような整数組が3つ存在するものは赤字で示しました。$87539319$はそうした整数のうちで最小であり、一般化されたタクシー数のうち $Ta(3)=87539319$ として知られています(この表記法によると $Ta(2)=1729$ となります)。

余談ですが、タクシー数 $Ta(2)=1729$ を与える立法数の組を求めさせる問題が2009年の一橋大学の入試問題として出題されています。

$N$ $a,b,c,d$
1729 1, 12, 9, 10
4104 2, 16, 9, 15
13832 2, 24, 18, 20
20683 10, 27, 19, 24
32832 4, 32, 18, 30
39312 2, 34, 15, 33
40033 9, 34, 16, 33
46683 3, 36, 27, 30
64232 17, 39, 26, 36
65728 12, 40, 31, 33
110656 4, 48, 36, 40
110808 6, 48, 27, 45
134379 12, 51, 38, 43
149389 8, 53, 29, 50
165464 20, 54, 38, 48
171288 17, 55, 24, 54
195841 9, 58, 22, 57
216027 3, 60, 22, 59
216125 5, 60, 45, 50
262656 8, 64, 36, 60
314496 4, 68, 30, 66
320264 18, 68, 32, 66
327763 30, 67, 51, 58
373464 6, 72, 54, 60
402597 42, 69, 56, 61
439101 5, 76, 48, 69
443889 17, 76, 38, 73
513000 10, 80, 45, 75
513856 34, 78, 52, 72
515375 15, 80, 54, 71
525824 24, 80, 62, 66
558441 30, 81, 57, 72
593047 7, 84, 63, 70
684019 51, 82, 64, 75
704977 2, 89, 41, 86
805688 11, 93, 30, 92
842751 23, 94, 63, 84
885248 8, 96, 72, 80
886464 12, 96, 54, 90
920673 20, 97, 33, 96
955016 24, 98, 63, 89
984067 35, 98, 59, 92
994688 29, 99, 60, 92
1009736 50, 96, 59, 93
1016496 47, 97, 66, 90
1061424 6, 102, 45, 99
1073375 23, 102, 60, 95
1075032 24, 102, 76, 86
1080891 27, 102, 48, 99
1092728 1, 103, 64, 94
1195112 16, 106, 58, 100
1260441 9, 108, 81, 90
1323712 40, 108, 76, 96
1331064 4, 110, 67, 101
1370304 34, 110, 48, 108
1407672 14, 112, 63, 105
1533357 62, 109, 90, 93
1566728 18, 116, 44, 114
1609272 55, 113, 92, 94
1728216 6, 120, 44, 118
1729000 10, 120, 90, 100
1734264 51, 117, 78, 108
1774656 36, 120, 93, 99
1845649 42, 121, 49, 120
2048391 2, 127, 95, 106
2101248 16, 128, 72, 120
2301299 11, 132, 99, 110
2418271 23, 134, 95, 116
2515968 8, 136, 60, 132
2562112 36, 136, 64, 132
2585375 50, 135, 95, 120
2622104 60, 134, 102, 116
2691451 18, 139, 94, 123
2864288 10, 142, 103, 121
2987712 12, 144, 108, 120
2991816 18, 144, 81, 135
3220776 84, 138, 112, 122
3242197 76, 141, 85, 138
3375001 1, 150, 73, 144
3375008 2, 150, 83, 141
3511872 4, 152, 41, 151
3512808 10, 152, 96, 138
3551112 34, 152, 76, 146
3587409 18, 153, 121, 122
3628233 36, 153, 114, 129
3798613 13, 156, 117, 130
3813992 26, 156, 123, 125
4033503 24, 159, 87, 150
4104000 20, 160, 90, 150
4110848 68, 156, 104, 144
4123000 30, 160, 108, 142
4174281 10, 161, 84, 153
4206592 48, 160, 124, 132
4342914 23, 163, 121, 137
4467528 60, 162, 114, 144
4505949 24, 165, 86, 157
4511808 27, 165, 100, 152
4607064 32, 166, 129, 135
4624776 51, 165, 72, 162
4673088 25, 167, 64, 164
4744376 14, 168, 126, 140
4914000 10, 170, 75, 165
4931101 76, 165, 102, 157
5004125 45, 170, 80, 165
5135104 36, 172, 93, 163
5287707 27, 174, 66, 171
5318677 37, 174, 136, 141
5446359 71, 172, 135, 144
5462424 22, 176, 99, 165
5472152 102, 164, 128, 150
5639816 4, 178, 82, 172
5772403 51, 178, 115, 162
5799339 40, 179, 107, 166
5832729 9, 180, 66, 177
5835375 15, 180, 135, 150
6013000 72, 178, 115, 165
6017193 57, 180, 113, 166
6058747 99, 172, 120, 163
6230504 10, 184, 67, 181
6434883 3, 186, 59, 184
6445504 22, 186, 60, 184
6517000 66, 184, 105, 175
6742008 46, 188, 126, 168
7081984 16, 192, 144, 160
7091712 24, 192, 108, 180
7094269 70, 189, 133, 168
7365384 40, 194, 66, 192
7620661 45, 196, 133, 174
7640128 48, 196, 126, 178
7872536 70, 196, 118, 184
7882245 93, 192, 149, 166
7957504 58, 198, 120, 184
8029000 85, 195, 130, 180
8077888 100, 192, 118, 186
8131968 94, 194, 132, 180
8216000 60, 200, 155, 165
8275176 32, 202, 129, 183
8491392 12, 204, 90, 198
8493039 15, 204, 79, 200
8494577 17, 204, 153, 170
8587000 46, 204, 120, 190
8600256 48, 204, 152, 172
8647128 54, 204, 96, 198
8741824 2, 206, 128, 188
8849601 90, 201, 153, 174
8872487 14, 207, 140, 183
9016488 26, 208, 117, 195
9261729 9, 210, 161, 172
9443761 68, 209, 146, 185
9560896 32, 212, 116, 200
10036845 72, 213, 109, 206
10083528 18, 216, 162, 180
10202696 50, 216, 167, 177
10424232 40, 218, 101, 211
10469816 81, 215, 120, 206
10589696 80, 216, 152, 192
10648512 8, 220, 134, 202
10702783 135, 202, 166, 183
10765603 64, 219, 163, 186
10811017 84, 217, 145, 198
10821896 116, 210, 147, 197
10870119 126, 207, 168, 183
10962432 68, 220, 96, 216
11097567 20, 223, 159, 192
11261376 28, 224, 126, 210
11478584 73, 223, 94, 220
11548089 17, 226, 54, 225
11557000 24, 226, 55, 225
11855727 15, 228, 144, 207
11859211 19, 228, 171, 190
11985003 51, 228, 114, 219
12266856 124, 218, 180, 186
12533824 36, 232, 88, 228
12650337 10, 233, 108, 225
12753160 47, 233, 176, 194
12874176 110, 226, 184, 188
13484016 14, 238, 105, 231
13489875 80, 235, 171, 204
13490533 21, 238, 114, 229
13623913 121, 228, 186, 193
13731319 63, 238, 112, 231
13825728 12, 240, 88, 236
13832000 20, 240, 180, 200
13851000 30, 240, 135, 225
13874112 102, 234, 156, 216
13915125 45, 240, 162, 213
14197248 72, 240, 186, 198
14482503 87, 240, 94, 239
14765192 84, 242, 98, 240
14916727 31, 246, 103, 240
14978061 45, 246, 173, 214
15077907 90, 243, 171, 216
15187041 156, 225, 169, 218
15201368 68, 246, 185, 207
15398936 80, 246, 179, 213
15438250 1, 249, 135, 235
15682680 119, 241, 168, 222
16012269 21, 252, 189, 210
16387128 4, 254, 190, 212
16387189 5, 254, 197, 206
16587207 18, 255, 151, 236
16776487 58, 255, 183, 220
16797375 60, 255, 190, 215
16809984 32, 256, 144, 240
16983854 21, 257, 167, 231
17045567 87, 254, 191, 216
17999072 146, 246, 207, 209
18406603 19, 264, 75, 262
18410392 22, 264, 198, 220
18468513 153, 246, 192, 225
18617625 20, 265, 118, 257
18673625 40, 265, 145, 250
19034379 6, 267, 123, 258
19250163 11, 268, 60, 267
19346168 46, 268, 190, 232
20072017 73, 270, 177, 244
20127744 16, 272, 120, 264
20162952 34, 272, 153, 255
20496896 72, 272, 128, 264
20616463 111, 268, 151, 258
20683000 100, 270, 190, 240
20976832 120, 268, 204, 232
21036743 23, 276, 207, 230
21125384 69, 275, 194, 240
21411000 85, 275, 120, 270
21531608 36, 278, 188, 246
21753576 33, 279, 90, 276
21994875 35, 280, 99, 276
22031576 119, 273, 182, 252
22426369 62, 281, 176, 257
22543417 49, 282, 130, 273
22544704 84, 280, 217, 231
22754277 69, 282, 189, 252
22914304 20, 284, 206, 242
23901696 24, 288, 216, 240
23934528 36, 288, 162, 270
24480125 45, 290, 110, 285
24666560 29, 291, 92, 288
24767171 50, 291, 123, 284
24858171 60, 291, 99, 288
24897817 9, 292, 201, 256
25164405 22, 293, 228, 237
25371171 90, 291, 166, 275
25766208 168, 276, 224, 244
25785432 72, 294, 189, 267
25937576 152, 282, 170, 276
26122131 107, 292, 235, 236
26506467 35, 298, 219, 252
26569809 105, 294, 177, 276
26856576 87, 297, 180, 276
27000008 2, 300, 146, 288
27000064 4, 300, 166, 282
27003375 15, 300, 110, 295
27015625 25, 300, 225, 250
27262872 150, 288, 177, 279
27445392 141, 291, 198, 270
27529073 110, 297, 209, 264
27831951 24, 303, 151, 290
27882127 40, 303, 234, 247
28094976 8, 304, 82, 302
28102464 20, 304, 192, 276
28149336 38, 304, 171, 285
28408896 68, 304, 152, 292
28658448 18, 306, 135, 297
28699272 36, 306, 242, 244
28703269 37, 306, 189, 280
28981125 69, 306, 180, 285
29025864 72, 306, 228, 258
29184057 81, 306, 144, 297
29503656 3, 309, 192, 282
29791125 5, 310, 66, 309
30388904 26, 312, 234, 260
30511936 52, 312, 246, 250
31256000 5, 315, 248, 252
32114143 159, 304, 223, 276
32268024 48, 318, 174, 300
32530680 41, 319, 72, 318
32832000 40, 320, 180, 300
32886784 136, 312, 208, 288
32973759 80, 319, 194, 295
32984000 60, 320, 216, 284
33108264 188, 298, 225, 279
33394248 20, 322, 168, 306
33652736 96, 320, 248, 264
34031907 27, 324, 243, 270
34743312 46, 326, 242, 274
35740224 120, 324, 228, 288
35938728 12, 330, 201, 303
36047592 48, 330, 172, 314
36050904 112, 326, 133, 323
36094464 54, 330, 200, 304
36856512 64, 332, 258, 270
36998208 102, 330, 144, 324
37384704 50, 334, 128, 328
37498032 62, 334, 243, 285
37955008 28, 336, 252, 280
38007144 42, 336, 189, 315
38253878 87, 335, 173, 321
39312000 20, 340, 150, 330
39448808 152, 330, 204, 314
39639691 88, 339, 150, 331
40033000 90, 340, 160, 330
40134584 94, 340, 157, 331
40183262 81, 341, 107, 339
40354119 8, 343, 191, 322
40519115 116, 339, 267, 278
40970375 150, 335, 255, 290
41080832 72, 344, 186, 326
41258737 58, 345, 144, 337
41301953 62, 345, 233, 306
41343640 86, 344, 197, 323
41400639 186, 327, 270, 279
41436873 72, 345, 268, 281
41511232 105, 343, 162, 334
41892515 48, 347, 243, 302
42154840 22, 348, 193, 327
42168581 29, 348, 261, 290
42276843 51, 348, 251, 298
42301656 54, 348, 132, 342
42549416 74, 348, 272, 282
43450344 165, 339, 276, 282
43570872 142, 344, 270, 288
43699392 44, 352, 198, 330
43777216 204, 328, 256, 300
43852536 62, 352, 243, 309
43946000 158, 342, 275, 285
43997625 22, 353, 250, 305
44409995 138, 347, 176, 339
45118528 8, 356, 164, 344
45440551 130, 351, 247, 312
45882739 3, 358, 115, 354
46091997 84, 357, 266, 301
46179224 102, 356, 230, 324
46343059 107, 356, 227, 326
46394712 80, 358, 214, 332
46661832 18, 360, 132, 354
46683000 30, 360, 270, 300
46825128 153, 351, 234, 324
47915712 108, 360, 279, 297
48104000 144, 356, 230, 330
48137544 114, 360, 226, 332
48316662 127, 359, 225, 333
48469976 198, 344, 240, 326
49832523 126, 363, 147, 360
49844032 20, 368, 134, 362
49849856 24, 368, 153, 359
49933368 46, 368, 207, 345
50324625 210, 345, 280, 305
51240427 56, 371, 203, 350
51278929 113, 368, 284, 305
51479064 6, 372, 118, 368
51508639 31, 372, 279, 310
51564032 44, 372, 120, 368
52136000 132, 368, 210, 350
52242463 199, 354, 208, 351
52324272 22, 374, 165, 363
52739288 17, 375, 251, 333
53226297 41, 376, 97, 374
53283923 99, 374, 176, 363
53936064 92, 376, 252, 336
54267003 186, 363, 212, 355
54269091 232, 347, 274, 323
54814509 93, 378, 301, 302
54887625 25, 380, 240, 345
55306557 6, 381, 285, 318
55486125 85, 380, 190, 365
56388339 179, 370, 272, 331
56655872 32, 384, 288, 320
56733696 48, 384, 216, 360
56754152 140, 378, 266, 336
57066633 2, 385, 252, 345
58438072 30, 388, 307, 309
58751784 119, 385, 168, 378
58923072 80, 388, 132, 384
59913728 125, 387, 216, 368
60271939 123, 388, 180, 379
60698521 4, 393, 282, 337
60965288 90, 392, 266, 348
61121024 96, 392, 252, 356
61555832 95, 393, 302, 324
61568208 74, 394, 291, 333
62108397 21, 396, 238, 365
62135073 33, 396, 297, 330
62980288 140, 392, 236, 368
63057960 186, 384, 298, 332
63660032 116, 396, 240, 368
64125000 50, 400, 225, 375
64232000 170, 390, 260, 360
64421875 75, 400, 270, 355
64623104 200, 384, 236, 372
65055744 188, 388, 264, 360
65272753 166, 393, 300, 337
65293317 69, 402, 285, 348
65573928 103, 401, 164, 394
65728000 120, 400, 310, 330
66201408 64, 404, 258, 366
66469429 34, 405, 237, 376
66763333 285, 352, 309, 334
67173463 63, 406, 154, 399
67931136 24, 408, 180, 396
67944312 30, 408, 158, 400
67956616 34, 408, 306, 340
68007673 222, 385, 241, 378
68696000 92, 408, 240, 380
68802048 96, 408, 304, 344
69177024 108, 408, 192, 396
69583995 54, 411, 187, 398
69625969 89, 410, 281, 362
69805125 150, 405, 285, 360
69934592 4, 412, 256, 376
70449093 16, 413, 245, 382
70796808 180, 402, 306, 348
70957971 3, 414, 323, 334
70979896 28, 414, 280, 366
71292312 150, 408, 283, 365
71509312 33, 415, 82, 414
72131904 52, 416, 234, 390
72669177 54, 417, 282, 369
73369088 95, 417, 176, 408
73819863 199, 404, 290, 367
73842713 110, 417, 236, 393
74093832 18, 420, 322, 344
74097261 21, 420, 154, 413
74130875 35, 420, 315, 350
75550088 136, 418, 292, 370
76487168 64, 424, 232, 400
77129864 113, 423, 320, 354
77335776 30, 426, 309, 363
77414967 120, 423, 271, 386
79598125 45, 430, 213, 412
79692193 57, 430, 313, 366
80102574 131, 427, 255, 399
80208576 151, 425, 332, 352
80294760 144, 426, 218, 412
80668224 36, 432, 324, 360
80779032 54, 432, 243, 405
81621568 100, 432, 334, 354
83393856 80, 436, 202, 422
83758528 162, 430, 240, 412
84266000 62, 438, 125, 435
84637287 32, 439, 166, 431
84717568 160, 432, 304, 384
85188096 16, 440, 268, 404
85492792 187, 429, 286, 396
85502375 255, 410, 320, 375
85622264 270, 404, 332, 366
86124824 128, 438, 326, 372
86316741 142, 437, 293, 394
86368464 26, 442, 195, 429
86488136 168, 434, 290, 396
86575168 232, 420, 294, 394
86960952 252, 414, 336, 366
87029432 45, 443, 318, 380
87483968 132, 440, 341, 363
87539319 167, 436, 228, 423
87539319 167, 436, 255, 414
87539319 228, 423, 255, 414
87579037 37, 444, 333, 370
87699456 136, 440, 192, 432
87952501 117, 442, 208, 429
88122125 10, 445, 205, 430
88129323 106, 443, 235, 422
88780536 40, 446, 318, 384
89576767 64, 447, 127, 444
90091008 56, 448, 252, 420
90527229 134, 445, 197, 436
91034307 160, 443, 332, 379
91125027 3, 450, 219, 432
91125216 6, 450, 249, 423
91140435 107, 448, 211, 434
91828672 146, 446, 188, 440
92384712 34, 452, 108, 450
92456000 48, 452, 110, 450
94100426 153, 449, 359, 363
94820544 12, 456, 123, 453
94845816 30, 456, 288, 414
94873688 38, 456, 342, 380
95547816 47, 457, 90, 456
95731489 66, 457, 97, 456
95880024 102, 456, 228, 438
96753187 147, 454, 283, 420
96860043 54, 459, 363, 366
97867441 81, 460, 145, 456
97962291 108, 459, 342, 387
98040969 89, 460, 321, 402
98134848 248, 436, 360, 372
99768753 105, 462, 328, 401
100092456 58, 464, 261, 435
100270592 72, 464, 176, 456
100711000 55, 465, 150, 460
101194704 2, 466, 165, 459
101202696 20, 466, 216, 450
101615579 170, 459, 323, 408
101700872 242, 444, 315, 413
102025280 94, 466, 352, 388
102562551 39, 468, 351, 390
102977784 78, 468, 369, 375
102993408 220, 452, 368, 376
104212017 73, 470, 337, 404
105343875 115, 470, 315, 420
105997327 239, 452, 296, 431
106243219 307, 426, 363, 388
106496767 7, 474, 255, 448
107872128 28, 476, 210, 462
107919000 160, 470, 342, 408
107924264 42, 476, 228, 458
108904581 72, 477, 261, 450
108991304 242, 456, 372, 386
109850552 126, 476, 224, 462
110087432 57, 479, 285, 443
110605824 24, 480, 176, 472
110656000 40, 480, 360, 400
110808000 60, 480, 270, 450
110992896 204, 468, 312, 432
111321000 90, 480, 324, 426
111485673 185, 472, 222, 465
111906333 150, 477, 229, 464
112422709 210, 469, 357, 406
112705587 30, 483, 252, 459
113577984 144, 480, 372, 396
114103808 27, 485, 152, 480
115084125 100, 485, 165, 480
115860024 174, 480, 188, 478
115942104 167, 481, 384, 390
116215603 11, 488, 203, 476
116773741 202, 477, 306, 445
117132750 145, 485, 303, 447
117258678 69, 489, 363, 411
118121536 168, 484, 196, 480
118129375 138, 487, 360, 415
118974960 251, 469, 378, 402
119082880 153, 487, 316, 444
119164409 41, 492, 369, 410
119333816 62, 492, 206, 480
119377000 120, 490, 315, 445
119793457 142, 489, 268, 465
119824488 11, 493, 90, 492
119824488 11, 493, 346, 428
119824488 90, 492, 346, 428
120623256 180, 486, 342, 432
121287376 1, 495, 334, 438
121496328 312, 450, 338, 436
121610944 136, 492, 370, 414
121660623 72, 495, 258, 471
121818816 81, 495, 300, 456
122262264 62, 496, 279, 465
122810129 36, 497, 345, 434
123008375 175, 490, 295, 460
123191488 160, 492, 358, 426
123262120 303, 457, 394, 396
123506000 2, 498, 270, 470
124336000 145, 495, 300, 460
124390728 96, 498, 387, 405
124868952 153, 495, 216, 486
125461440 238, 482, 336, 444
126173376 75, 501, 192, 492
126217000 250, 480, 295, 465
126516061 165, 496, 262, 477
126906624 124, 500, 307, 461
127062000 235, 485, 330, 450
127396178 51, 503, 237, 485
127525671 64, 503, 159, 498
128098152 42, 504, 378, 420
128966383 135, 502, 354, 439
129333078 153, 501, 311, 463
129380329 84, 505, 352, 441
131097024 8, 508, 380, 424
131097512 10, 508, 394, 412
131126303 31, 508, 250, 487
131997229 50, 509, 341, 452
132678000 30, 510, 225, 495
132697656 36, 510, 302, 472
133139727 228, 495, 306, 471
134171875 115, 510, 300, 475
134211896 116, 510, 366, 440
134379000 120, 510, 380, 430
134479872 64, 512, 288, 480
135111375 135, 510, 240, 495
135870832 42, 514, 334, 462
136004184 137, 511, 294, 480
136364536 174, 508, 382, 432
136591000 5, 515, 320, 470
137467603 43, 516, 387, 430
137854899 243, 498, 395, 424
138090771 294, 483, 392, 427
138647808 108, 516, 279, 489
141117704 221, 507, 338, 468
141864697 190, 513, 361, 456
142098327 132, 519, 199, 512
142244648 20, 522, 381, 443
142768089 81, 522, 198, 513
143604279 111, 522, 359, 460
143604279 111, 522, 408, 423
143604279 359, 460, 408, 423
143992576 292, 492, 414, 418
144370863 79, 524, 356, 463
144404416 156, 520, 403, 429
145461771 134, 523, 352, 467
145908847 346, 471, 360, 463
146548376 57, 527, 275, 501
147051693 213, 516, 405, 432
147187369 225, 514, 264, 505
147252824 38, 528, 150, 524
147283136 44, 528, 396, 440
147485448 66, 528, 297, 495
147748104 306, 492, 384, 450
148941000 40, 530, 236, 514
149389000 80, 530, 290, 500
149496256 264, 508, 369, 463
150611643 35, 532, 336, 483
151212412 147, 529, 261, 511
152253927 119, 532, 266, 511
152275032 12, 534, 246, 516
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154001304 22, 536, 120, 534
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