数字串的通项公式

王守恩

A003348——Numbers that are the sum of 3 positive 5th powers.——A344641是同一个数字串。

3, 34, 65, 96, 245, 276, 307, 487, 518, 729, 1026, 1057, 1088, 1268, 1299, 1510, 2049, 2080, 2291, 3072, 3127, 3158, 3189, 3369, 3400, 3611, 4150, 4181, 4392, 5173, 6251, 6282, 6493, 7274, 7778, 7809, 7840, 8020, 8051, 8262, 8801, 8832,

Union[n /. Solve[{a^5 + b^5 + c^5 == n, 354810000 > n > 354000000, a >= b >= c > 0}, {n, a, b, c}, Integers]]——用我们的公式也可以出来前20000项。

{353509243, 353522298, 353524961, 353525699, 353528450, 353708774, 353730794, 353747250, 353798400, 353864576, 353937693, 353987492, 354010368, 354015599, 354017056, 354019995, 354036451, 354111492, 354132448,
354262984, 354295305, 354354750, 354391276, 354405537, 354407732, 354483062, 354542101, 354550285, 354597981, 354661594, 354790877, 354790908, 354791119, 354791900, 354793792, 354794001, 354798652, 354807683}

19963 353509243
19964 353522298
19965 353524961
19966 353525699
19967 353528450
19968 353708774
19969 353730794
19970 353747250
19971 353798400
19972 353864576
19973 353937693
19974 353987492
19975 354010368
19976 354015599
19977 354017056
19978 354019995
19979 354036451
19980 354111492
19981 354132448
19982 354262984
19983 354295305
19984 354354750
19985 354391276
19986 354405537
19987 354407732
19988 354483062
19989 354542101
19990 354550285
19991 354597981
19992 354661594
19993 354790877
19994 354790908
19995 354791119
19996 354791900
19997 354793792
19998 354794001
19999 354798652
20000 354807683

王守恩

A046080——\(\D n^2 = a_{1}^2 + a_{2}^2\)
0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 4, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 1, 1, 1, 0, 0, 0,

A181786——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2\)
0, 0, 0, 1, 0, 0, 1, 1, 0, 3, 0, 2, 1, 1, 1, 3, 0, 2, 3, 3, 0, 6, 2, 3, 1, 2, 1, 8, 1, 3, 3, 4, 0, 10, 2, 5, 3, 4, 3, 8, 0, 5, 6, 6, 2, 11, 3, 6, 1, 8, 2, 12, 1, 6, 8, 8, 1, 15, 3, 8, 3, 7, 4, 20, 0, 6, 10, 9, 2, 16, 5, 9, 3, 9, 4, 15, 3, 15, 8, 10, 0, 22, 5, 11, 6, 9, 6, 18,

OEIS没有——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2\)
{0, 1, 0, 1, 1, 2, 2, 1, 2, 5, 3, 2, 5, 8, 9, 1, 7, 10, 9, 5, 16, 12, 13, 2, 19, 18, 22, 8, 20, 32, 23, 1, 35, 25, 42, 10, 32, 31, 51, 5, 38, 55, 42, 12, 80, 43, 50, 2, 63, 62, 83, 18, 63, 75, 91, 8, 103, 65, 77, 32, 83, 74, 144, 1, 127, 116, 99, 25, 151, 133}

A179015——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2 +a_{5}^2\)
0, 0, 0, 1, 1, 1, 2, 5, 2, 6, 6, 9, 9, 15, 8, 25, 20, 21, 25, 39, 26, 46, 44, 57, 49, 71, 52,102, 81, 81, 99, 145,92, 156, 126,164, 160, 204, 151, 247, 217, 236, 245, 326, 211, 357, 319, 381, 360, 416, 344, 518, 446, 476, 450, 670, 468, 675, 607, 661,

OEIS没有——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2 +a_{5}^2 +a_{6}^2\)
{0, 0, 1, 0, 1, 4, 3, 2, 10, 10, 12, 16, 20, 28, 40, 32, 46, 81, 68, 84, 107, 138, 131, 160, 184, 250, 253, 272, 296, 459, 374, 452, 501, 650, 599, 732, 714,981, 901, 1124, 1036, 1405, 1236, 1566, 1558, 1994, 1718, 2086, 1961, 2873, 2324, 3004}

OEIS没有——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2 +a_{5}^2 +a_{6}^2 +a_{7}^2 \)
{0, 0, 0, 1, 2, 1, 4, 7, 9, 13, 19, 29, 35, 46, 68, 87, 101, 138, 159, 212, 261, 308, 348, 505, 488, 633, 756, 905, 936, 1268, 1250, 1647, 1808, 2113, 2078, 3022, 2726, 3508, 3755, 4400, 4320, 5956, 5357, 6984, 7025, 8427, 8015, 11241, 9659}

OEIS没有——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2 +a_{5}^2 +a_{6}^2+a_{7}^2  +a_{8}^2 \)
{0, 0, 0, 1, 1, 1, 5, 9, 9, 15, 30, 32, 61, 70, 93, 149, 202, 198, 338, 421, 486, 608, 845, 942, 1280, 1423, 1719, 2246,  2673, 2838, 3752,  4478, 4869, 5736, 7094, 7912, 9356, 10363, 11844, 14715, 16045, 17316, 20649, 24188, 25767, 29132}

OEIS没有——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2 +a_{5}^2 +a_{6}^2 +a_{7}^2 +a_{8}^2 +a_{9}^2 \)
{0, 0, 1, 0, 1, 4, 3, 5, 16, 19, 27, 52, 63, 95, 155, 182, 260, 389, 479, 626, 880, 1088,1401, 1792, 2265, 2811, 3599, 4275,5386, 6507, 7998,9398, 11620, 13690, 16545, 19027, 23217, 26841, 31677, 36210, 43529, 48992, 58411, 65608, 75993}

可以有统一的通项公式——Count[PowersRepresentations[#^2, 9, 2], w_ /; (Times@@w) > 0] & /@Range[50]

王守恩

OEIS没有——\(\D n=(a+b)(a^2+b^2)\)——{1, 4, 5, 8, 13, 15, 20, 25, 27, 32, 40, 41, 51, 61, 64, 65, 68, 85, 87, 104, 108, 113, 120, 125, 135, 145, 148, 156, 160, 175, 181, 185, 195, 200, 203, 216, 221, 232, 256, 259, 260, 265, 267, 272,

OEIS没有——\(\D n=(a+b)(a^3+b^3)\)——{1, 4, 7, 16, 19, 27, 37, 52, 61, 64, 81, 91, 112, 127, 169, 175, 189, 196, 217, 256, 271, 304, 324, 325, 331, 351, 397, 432, 436, 469, 496, 547, 567, 592, 625, 631, 637, 721, 756, 772}

OEIS没有——\(\D n=(a+b)(a^4+b^4)\)——{1, 4, 17, 32, 51, 97, 128, 164, 243, 328, 337, 485, 544, 771, 881, 972, 1024, 1285, 1412, 1632, 1921, 1923, 2359}

OEIS没有——\(\D n=(a^2+b^2)(a^3+b^3)\)——{1, 4, 32, 35, 45, 128, 243, 247, 260, 280, 455, 925, 972, 1024, 1071, 1105, 1120, 1440}

OEIS没有——\(\D n=(a+b)(a^5+b^5)\)——{1, 4, 31, 64, 99, 211, 256, 484, 729, 781, 976, 1375, 1984, 2101, 2916, 3069}

OEIS没有——\(\D n=(a^2+b^2)(a^4+b^4)\)——{1, 4, 64, 85, 256, 729, 820, 1261, 2916, 4096, 4369, 5440}

OEIS没有——\(\D n=(a+b)(a^6+b^6)\)——{1, 4, 65, 128, 195, 512, 793, 1460, 2187, 2920, 3965, 4825}

OEIS没有——\(\D n=(a^2+b^2)(a^5+b^5)\)——{1, 4, 128, 155, 165, 512, 2187, 2420, 2440, 2743, 3575}

OEIS没有——\(\D n=(a^3+b^3)(a^4+b^4)\)——{1, 4, 119, 128, 153, 512, 1843, 2132, 2187, 2296, 3395}

可以有统一的通项公式——Select[Range[2500], FindInstance[(a^3 + b^3) (a^4 + b^4) == #, {a, b}, Integers,1] ≠ {} &]——这通项公式有点慢。

王守恩

\(\D n=(a+b+c)(a^3+b^3+c^3)\)

1{{a -> 0, b -> 0, c -> -1}}, {}, {},
4{{a -> 1, b -> 1, c -> 0}}, {}, {},
7{{a -> 1, b -> 0, c -> -2}}, {},
9{{a -> 1, b -> 1, c -> 1}}, {}, {}, {}, {}, {}, {},
16{{a -> 1, b -> -1, c -> -2}}, {}, {},
19{{a -> 3, b -> 0, c -> -2}}, {}, {}, {}, {}, {},
25{{a -> 3, b -> -1, c -> -1}}, {},
27{{a -> 0, b -> -1, c -> -2}}, {}, {}, {}, {}, {}, {}, {}, {}, {},
37{{a -> 3, b -> 0, c -> -4}}, {}, {},
40{{a -> -1, b -> -1, c -> -2}}, {}, {}, {}, {},
45{{a -> 1, b -> -2, c -> -2}}, {}, {}, {}, {}, {}, {},
52{{a -> 3, b -> 0, c -> -1}}, {}, {},
55{{a -> 2, b -> 1, c -> -4}}, {}, {}, {}, {}, {},
61{{a -> 5, b -> 0, c -> -4}}, {}, {},
64{{a -> 0, b -> -2, c -> -2}}, {}, {},
67{{a -> 144, b -> 73, c -> -150}, {}, {}, {}, {},
72{{a -> 11, b -> -7, c -> -10}}, {}, {}, {},
76{{a -> 3, b -> -1,  c -> -4}}, {}, {}, {}, {},
81{{a -> 3, b -> 1, c -> -1}}, {}, {}, {},
85{{a -> -1, b -> -2, c -> -2}}, {}, {}, {}, {}, {},
91{{a -> 6, b -> 0, c -> -5}}, {}, {}, {}, {}, {},
97{{a -> 3, b -> 1, c -> -5}}, {}, {}, {}}......
{1, 4, 7, 9, 16, 19, 25, 27, 37, 40, 45, 52, 55, 61, 64, 67, 72, 76, 81, 85, 91, 97, ——22
109, 112, 121, 124, 127, 135, 136, 144, 145, 151, 169, 171, 175, 181, 184, 189, 196,——17
207, 216, 217, 232, 235, 256, 265, 271, 277, 280, 289, 295, 297, ——13
301, 304, 315, 319, 324, 325, 331, 340, 351, 352, 355, 361, 369, 376, 379, 385, 391, 396, 397, ——19
400, 405, 415, 421, 424, 432, 436, 445, 459, 469, 472, 475, 487, 496, ——14
505, 511, 531, 532, 540, 544, 547, 556, 565, 567, 577, 592,——12
601, 616, 621, 625, 631, 637, 640, 649, 657, 664, 675, ——11
700, 720, 721, 727, 729, 736, 756, 757, 760, 772, 775, 781, 784, 792, 796, ——15
805, 811, 817, 832, 837, 856, 864, 865, 880, 889, 891, 895,——12
904, 916, 919, 931, 940, 945, 955, 961, 972, 976, 981, 985, 991, ——13=148
1000, 1015, 1017, 1024, 1027, 1044, 1045, 1051, 1057, 1072, 1075, 1080, 1081, 1096, ——14
1111, 1116, 1120, 1141, 1147, 1152, 1156, 1161, 1165, 1177, 1180, 1189, 1192, 1197, ——14
1204, 1215, 1216, 1225, 1240, 1261, 1264, 1267, 1269, 1276, 1296, ——11
1312, 1321, 1336, 1341, 1351, 1360, 1377, 1387, ——8
1405, 1411, 1431, 1432, 1435, 1441, 1444, 1456, 1467, 1485, ——10
1504, 1512, 1519, 1537, 1539, 1540, 1552, 1561, 1576, 1585, ——10
1600, 1615, 1620, 1621, 1636, 1639, 1645, 1647, 1657, 1665, 1675, 1687, 1692, 1696, ——14
1701, 1705, 1717, 1729, 1732, 1735, 1744, 1764, 1765, 1771, 1792, ——11
1801, 1809, 1825, 1836, 1840, 1872, 1891, ——7
1900, 1917, 1927, 1936, 1939, 1944, 1945, 1951, 1960, 1969, 1971, 1975, 1984, 1996, ——14=113
2016, 2017, 2025, 2032, 2052, 2056, 2065, 2080, ——8
2107, 2149, 2160, 2161, 2167, 2176, 2185, 2187, ——8
2209, 2236, 2241, 2245, 2269, 2284, 2296, 2299, ——8
2304, 2305, 2311, 2320, 2341, 2349, 2356, 2376, 2392, ——9
2401, 2416, 2425, 2431, 2437, 2440, 2449, 2457, 2475, 2484, ——10
2500, 2511, 2512, 2515, 2521, 2524, 2527, 2560, 2575, 2587, 2592, 2596, ——12
2601, 2605, 2611, 2632, 2635, 2647, 2656, 2665, 2689, 2695, ——10
2700, 2701, 2704, 2716, 2727, 2736, 2737, 2752, 2755, 2761, 2776, 2781, 2791, ——12
2800, 2824, 2835, 2884, 2896, ——5
2905, 2916, 2920, 2944, 2947, 2952, 2956, 2961, 2971, 2977, 2980, 2992, 2997,——13 =93
3019, 3024, 3025, 3031, 3040, 3051, 3055, 3060, 3061, 3067, 3076, 3079, 3091, 3096, ——14
3105, 3121, 3132, 3136, 3145, 3151, 3157, 3159, 3169, 3175, 3184, ——12
3232, 3240, 3241, 3280, 3289, 3292, ——6
3312, 3316, 3321, 3325, 3367, 3376, ——6
3400, 3409, 3421, 3447, 3451, 3456, 3457, 3472, 3475, 3496, ——10
3505, 3520, 3535, 3537, 3564, 3565, 3571, 3577, 3580, 3591, ——10
3601, 3604, 3609, 3616, 3645, 3664, 3672, 3697, ——8
3700, 3712, 3724, 3751, 3760, 3769, 3775, 3781, 3784, 3796,——10
3807, 3829, 3841, 3877, 3889, 3892, ——6
3904, 3925, 3949, 3952, 3969, 3976, 3979, 3991, 3996, 3997, ——10=90
4000, 4015, 4032, 4039, 4051, 4057, 4059, 4060, 4072, 4077, 4081, 4096, ——12
4104, 4120, 4131, 4135, 4141, 4159, 4176, 4185, ——8
4212, 4216, 4219, 4225, 4239, 4240, 4275, 4285, 4291, ——9
4320, 4321, 4336, 4345, 4375, 4396, ——6
4401, 4405, 4417, 4432, 4447, 4455, 4456, 4465, 4477, 4480, 4489, ——11
4536, 4537, 4540, 4552, 4555, 4561, 4581, 4585, 4591, ——9
4600, 4617, 4624, 4671, 4672, 4675, 4681, 4696, ——8
4705, 4711, 4720, 4725, 4741, 4744, 4752, 4779, 4789, 4792, 4795, ——11
4804, 4816, 4825, 4864, 4876, 4896, ——6
4900, 4912, 4915, 4921, 4941, 4945, 4960, 4969, 4995, ——9=89
5031, 5035, 5040, 5047, 5049, 5071, 5076, 5089, ——8
5101, 5104, 5125, 5131, 5137, 5140, 5152, 5157, 5161, 5164, 5167, 5184, 5185, ——13
5200, 5211, 5215, 5224, 5229, 5256, 5281, 5296, ——8
5320, 5341, 5344, 5347, 5380, 5392, 5395, ——7
5401, 5416, 5419, 5425, 5427, 5440, 5455, 5476, 5481, 5491, ——10
5500, 5524, 5536, 5572, 5575, 5580, 5584, 5587, ——8
5616, 5625, 5629, 5632, 5652, 5656, 5659, 5671, 5677, 5680, 5697, -——11
5731, 5751, 5761, 5767, 5776, 5797, ——6
5812, 5824, 5832, 5841, 5851, 5872, 5877, 5896, ——8
5904, 5920, 5929, 5941, 5956, 5971, 5976, ——7=86
6016, 6021, 6025, 6049, 6061, 6064, 6075, 6076, 6085, 6097, ——10
6100, 6129, 6136, 6145, 6156, 6157, 6160, 6172, 6175, 6181, 6184, ——11
6211, 6220, 6256, 6264, 6271, 6292, 6295, ——7
6301, 6304, 6316, 6319, 6325, 6336, 6337, 6340, 6345, 6352, 6355, 6385,——12
6400, 6417, 6427, 6457, 6460, 6480, 6484, 6487, 6496, ——9
6507, 6535, 6544, 6559, 6561, 6592, ——6
6615, 6625, 6631, 6640, 6664, 6696, ——6
6721, 6736, 6745, 6769, 6784, ——5
6805, 6811, 6820, 6831, 6844, 6856, 6880, 6885, 6895, ——9
6901, 6912, 6916, 6919, 6925, 6937, 6952, 6976, ——8=82
7000, 7011, 7021, 7036, 7047, 7057, 7065, ——7
7101, 7105, 7111, 7120, 7129, 7144, ——6
7201, 7209, 7216, 7231, 7236, 7249, 7255, 7261, 7267, ——9
7300, 7309, 7312, 7335, 7336, 7344, 7345, 7351, 7360, 7371, 7384, ——11
7425, 7441, 7452, 7456, 7479, 7480, 7489, ——7
7501, 7504, 7552, 7555, 7561, 7564, 7576, 7587, 7596, ——9
7600, 7632, 7651, 7660, ——4
7711, 7744, 7776, 7777, 7792,——5
7816, 7825, 7831, 7840, 7855, 7857, 7861, 7867, 7875, ——9
7900, 7912, 7915, 7924, 7936, 7939, 7945, 7947, 7957, 7975, 7984, 7987, 7996, ——13=80
8001, 8032, 8041, 8055, 8056, 8080, ——6
8100, 8101, 8107, 8116, 8125, 8136, 8145, 8176, 8181, 8185, 8191, ——11
8200, 8215, 8227, 8251, 8269, 8271, 8275, 8281, 8289, ——9
8305, 8316, 8317, 8332, 8371, 8397, ——6
8416, 8425, 8440, 8449, 8451, 8496, ——6
8505, 8512, 8536, 8560, 8587, ——5
8605, 8620, 8635, 8640, 8641, 8656, 8665, 8667, 8671, 8680, 8695, ——11
8701, 8704, 8721, 8725, 8737, 8752, 8775, ——7
8815, 8827, 8829, 8856, 8869, 8881, 8896, 8899,——8
8901, 8905, 8911, 8929, 8932, 8937, 8955, 8971, 8977, 8980, 8991, 8992, ——12=76
9000, 9040, 9045, 9072, 9097, ——5
9100, 9121, 9124, 9127, 9136, 9180, 9184, 9196, ——8
9205, 9207, 9232, 9241, 9261, 9265, 9280, 9295, ——8
9301, 9315, 9325, 9340, 9361, 9364, 9376, 9385, 9396, 9397,——10
9409, 9412, 9424, 9439, 9451, 9460, 9469, 9472, 9496, 9499, ——10
9504, 9505, 9517, 9520, 9531, 9535, 9576, 9577, 9592, ——9
9601, 9604, 9612, 9616, 9639, 9640, 9675, 9676, 9685, 9691, ——10
9712, 9720, 9721, 9724, 9751, ——5
9801, 9832, 9841, 9844, 9856, 9864, 9865, 9889, ——8
9909, 9919, 9936, 9937, 9952, 9955, 9976, ——7=80
10000}
代码A——Select[Range[10000], FindInstance[(a + b + c) (a^3 + b^3 + c^3) == #, {a, b, c}, Integers, 1] != {} &]

代码B——Union[Select[Sort[Flatten[Table[(a + b + c) (a^3 + b^3 + c^3), {n, 1544, 1544}, {a, -n, n}, {b, a, n}, {c, b, n}]]], 0 < # < 10000 &]]

代码A还是比代码B速度快一些？  361{a -> 1544, b -> -368, c -> -1537}

王守恩

A023042——Numbers whose cube is the sum of three distinct nonnegative cubes.——2025年5月7日

代码A——Select[Range[400], Length[PowersRepresentations[#^3, 3, 3]] > 1 &]
{6, 9, 12, 18, 19, 20, 24, 25, 27, 28, 29, 30, 36, 38, 40, 41, 42, 44, 45, 46, 48, 50, 53, 54, 56, 57, 58, 60, 63, 66, 67, 69, 70, 71, 72, 75, 76, 78, 80, 81, 82, 84, 85, 87, 88, 89, 90, 92, 93, 95, 96, 97, 99, 100, 102, 103, 105, 106, 108,
110, 111, 112, 113, 114, 115, 116, 117, 120, 121, 122, 123, 125, 126, 127, 129, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 144, 145, 147, 150, 151, 152, 153, 156, 159, 160, 162, 164, 167, 168, 170, 171, 172, 174, 175, 176,
177, 178, 179, 180, 181, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 209, 210, 212, 213, 214, 216, 217, 218, 219, 220, 222, 224, 225, 226, 228, 229, 230, 231,
232, 234, 235, 238, 239, 240, 241, 242, 243, 244, 246, 247, 249, 250, 251, 252, 254, 255, 256, 258, 260, 261, 262, 264, 265, 266, 267, 268, 269, 270, 274, 275, 276, 278, 279, 280, 281, 282, 284, 285, 287, 288, 289, 290, 291, 292,
293, 294, 295, 297, 298, 300, 302, 303, 304, 305, 306, 307, 308, 309, 311, 312, 315, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 330, 331, 332, 333, 334, 335, 336, 339, 340, 342, 344, 345, 346, 348, 349, 350, 351,
352, 354, 355, 356, 358, 360, 361, 362, 363, 364, 365, 366, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 398, 399, 400}

代码A——至少有两种三立方和表示 “=” 代码B——存在一组三个互不相同正整数的三立方和表示。

{6, 9, 12, 18, 19, 20, 24, 25, 27, 28, 29, 30, 36, 38, 40, 41, 42, 44, 45, 46, 48, 50, 53, 54, 56, 57, 58, 60, 63, 66, 67, 69, 70, 71, 72, 75, 76, 78, 80, 81, 82, 84, 85, 87, 88, 89, 90, 92, 93, 95, 96, 97, 99, 100, 102, 103, 105, 106, 108,
110, 111, 112, 113, 114, 115, 116, 117, 120, 121, 122, 123, 125, 126, 127, 129, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 144, 145, 147, 150, 151, 152, 153, 156, 159, 160, 162, 164, 167, 168, 170, 171, 172, 174, 175, 176,
177, 178, 179, 180, 181, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 209, 210, 212, 213, 214, 216, 217, 218, 219, 220, 222, 224, 225, 226, 228, 229, 230, 231,
232, 234, 235, 238, 239, 240, 241, 242, 243, 244, 246, 247, 249, 250, 251, 252, 254, 255, 256, 258, 260, 261, 262, 264, 265, 266, 267, 268, 269, 270, 274, 275, 276, 278, 279, 280, 281, 282, 284, 285, 287, 288, 289, 290, 291, 292,
293, 294, 295, 297, 298, 300, 302, 303, 304, 305, 306, 307, 308, 309, 311, 312, 315, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 330, 331, 332, 333, 334, 335, 336, 339, 340, 342, 344, 345, 346, 348, 349, 350, 351,
352, 354, 355, 356, 358, 360, 361, 362, 363, 364, 365, 366, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 398, 399, 400}
代码B——Select[Range[400], FindInstance[{a^3 + b^3 + c^3 == #^3, a > b > c > 0}, {a, b, c}, Integers, 1] != {} &]

王守恩

接楼上。

\(0^3=1^3-1^3-0^3\)
\(1^3=9^3-6^3-8^3\)
\(2^3=41^3-17^3-40^3\)
\(3^3=115^3-34^3-114^3\)
\(4^3=249^3-57^3-248^3\)
\(5^3=461^3-86^3-460^3\)
\(6^3=769^3-121^3-768^3\)
\(7^3=1191^3-162^3-1190^3\)
\(8^3=1745^3-209^3-1744^3\)
\(9^3=2449^3-262^3-2448^3\)
\(10^3=3321^3-321^3-3320^3\)
\(11^3=4379^3-386^3-4378^3\)
\(12^3=5641^3-457^3-5640^3\)
\(13^3=7125^3-534^3-7124^3\)
\(14^3=8849^3-617^3-8848^3\)
\(\cdots\cdots\)
把中间的数提出来——1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321, 386, 457, 534, 617, 706, 801, 902, 1009, 1122, 1241, 1366, 1497, 1634, 1777, 1926, 2081, 2242, 2409, 2582, 2761, 2946,——A056109——没有这条文。

王守恩

\(0^3=-1^3+1^3-0^3\)
\(1^3=1^3+2^3-2^3\)
\(2^3=15^3+9^3-16^3\)
\(3^3=59^3+22^3-60^3\)
\(4^3=151^3+41^3-152^3\)
\(5^3=309^3+66^3-310^3\)
\(6^3=551^3+97^3-552^3\)
\(7^3=895^3+134^3-896^3\)
\(8^3=1359^3+177^3-1360^3\)
\(9^3=1961^3+226^3-1962^3\)
\(10^3=2719^3+281^3-2720^3\)
\(11^3=3651^3+342^3-3652^3\)
\(12^3=4775^3+409^3-4776^3\)
\(13^3=6109^3+482^3-6110^3\)
\(14^3=7671^3+561^3-7672^3\)
\(15^3=9479^3+646^3-9480^3\)
\(\cdots\cdots\)
把中间的数提出来——1, 2, 9, 22, 41, 66, 97, 134, 177, 226, 281, 342, 409, 482, 561, 646, 737, 834, 937, 1046, 1161, 1282, 1409, 1542, 1681, 1826, 1977, 2134, 2297, 2466, 2641, 2822,—— A056105——没有这条文。

王守恩

A230562_Smallest number that is the sum of 2 positive 4th powers in >= n ways.——Oct 25 2013.
2, 635318657,
a(1) = 2 = 1^4 + 1^4.
a(2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.

A046881——Smallest number that is sum of 2 positive distinct n-th powers in 2 different ways.——Nov 02 2020.
5, 65, 1729, 635318657,
a(1) = 5 = 1^1 + 4^1 = 2^1 + 3^1.
a(2) = 65 = 1^2 + 8^2 = 4^2 + 7^2.
a(3) = 1729 = 1^3 + 12^3 = 9^3 + 10^3.
a(4) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.

A016078——Smallest number that is sum of 2 positive n-th powers in 2 different ways.Apr. ——03 2021.
4, 50, 1729, 635318657,
a(1) = 4 = 1^1 + 3^1 = 2^1 + 2^1.
a(2) = 50 = 1^2 + 7^2 = 5^2 + 5^2.
a(3) = 1729 = 1^3 + 12^3 = 9^3 + 10^3.
a(4) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.

A338800——Smallest number that is the sum of two distinct n-th powers of primes in two different ways.——Jul 01 2024.
16, 410, 6058655748, 3262811042,
a(1) = 16 = 3^1 + 13^1 = 5^1 + 11^1.
a(2) = 410 = 7^2 + 19^2 = 11^2 + 17^2.
a(3) = 6058655748 = 61^3 + 1823^3 = 1049^3 + 1699^3.
a(4) = 3262811042 = 7^4 + 239^4 = 157^4 + 227^4.

A374418——a(n) is the smallest number which can be represented as the sum of 2 distinct positive n-th powers in exactly 3 ways, or -1 if no such number exists.——Jul 08 2024.
7, 325, 87539319,
a(1) = 7=1^1 + 6^1 = 2^1 + 5^1 = 3^1 + 4^1.
a(2) = 325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2.
a(3) = 87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3.

王守恩

2026 \(\D=U_{1}^2+V_{1}^1-W_{1}^2=U_{2}^2+V_{2}^2-W_{2}^2=U_{3}^2+V_{3}^3-W_{3}^2=\cdots=U_{i}^2+V_{i}^i-W_{i}^2=\cdots=U_{2026}^2+V_{2026}^{2026}-W_{2026}^2\)

说明:  \(U_{i},V_{i},W_{i}\) = 互不相同正整数。

王守恩

A004437——Numbers that are not the sum of 4 distinct squares.4个不同整数平方和无解的数字。
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 31, 32, 33, 34, 36, 37, 40, 43, 44, 47, 48, 52, 55, 58, 60, 64, 67, 68, 72, 73, 76, 80, 82, 88, 92, 96, 97, 100, 103, 108, 112,

A000534——Numbers that are not the sum of 4 nonzero squares.4个非零整数平方和无解的数字。
1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41, 56, 96, 128, 224, 384, 512, 896, 1536, 2048, 3584, 6144, 8192, 14336, 24576, 32768, 57344, 98304, 131072, 229376, 393216, 524288, 917504, 1572864, 2097152,

A004441——Numbers that are not the sum of 4 distinct nonzero squares.4个不同非零整数不同平方和无解的数字。
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 52, 53, 55, 56, 58, 59, 60, 61, 64, 67, 68,

4个>1整数平方和无解的数字。{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 30, 32, 34, 35, 39, 41, 44, 46, 51, 55, 56, 62, 67, 89, 96,

4个不同>1整数平方和无解的数字{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,

4个>2整数平方和无解的数字。{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41,

A123069——Odd positive integers that are not the sum of four positive squares.4个奇正整数平方和无解的数字。
1, 3, 5, 9, 11, 17, 29, 41——sm4=Union[Total/@Tuples[Range[10]^2, 4]]; Select[Range[1, 100, 2], !MemberQ[sm4, #]&] (* James C. McMahon, Nov 15 2024 *)

A047701——All positive numbers that are not the sum of 5 nonzero squares.5个>0整数平方和无解的数字。
1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33——list; graph; refs; listen; history; text; internal format)

A004438 Numbers that are not the sum of 5 distinct squares. 不可表示为 5 个不同平方和的数。
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 52, 53, 56, 58, 59, 60, 61, 64, 67, 68, 69, 72, 73, 76, 77, 80, 83,

A123120——Numbers k>0 such that m+k is not the sum of m nonzero squares for any m>5.使得 m(m>5)个非零平方和无解的数字。
1, 2, 4, 5, 7, 10, 13——没看懂。

j[{a_, b_, c_, d_, e_}] := If[0 <= a < b < c < d < e, a^2 + b^2 + c^2 + d^2 + e^2, False]; Union[Complement[Range[500], DeleteCases[Flatten[Map[j, PowersRepresentations[#, 5, 2]] & /@ Range[500]], False]]]