累乗数
累乗数に関する性質 > 累乗和

自然数の累乗和  \sum _{k=1}^{n}k^{m}=1^{m}+2^{m}+3^{m}+\dotsb +n^{m}
| m | 数 | OEIS |
| --- | --- | --- |
| 1 | 三角数を参照 | A000217 |
| 2 | 四角錐数を参照 | A000330 |
| 3 | 立方数を参照 | A000537 |
| 4 | 1, 17, 98, 354, 979, 2275, 4676, 8772, 15333, … | A000538 |
| 5 | 1, 33, 276, 1300, 4425, 12201, 29008, 61776, … | A000539 |
| 6 | 1, 65, 794, 4890, 20515, 67171, 184820, 446964, … | A000540 |
| 7 | 1, 129, 2316, 18700, 96825, 376761, 1200304, 3297456, … | A000541 |
| 8 | 1, 257, 6818, 72354, 462979, 2142595, 7907396, 24684612, … | A000542 |
| 形 | 数 | OEIS |
| --- | --- | --- |
| 1n + 2n + 3n | 3, 6, 14, 36, 98, 276, 794, 2316, 6818, … | A001550 |
| 1n + 2n + 3n + 4n | 4, 10, 30, 100, 354, 1300, 4890, 18700, … | A001551 |
| 1n + 2n + 3n + 4n + 5n | 5, 15, 55, 225, 979, 4425, 20515, 96825, … | A001552 |
| 1n + 2n + 3n + ⋯ + 6n | 6, 21, 91, 441, 2275, 12201, 67171, 376761, … | A001553 |
| 1n + 2n + 3n + ⋯ + 7n | 7, 28, 140, 784, 4676, 29008, 184820, 1200304, … | A001554 |
| 1n + 2n + 3n + ⋯ + 8n | 8, 36, 204, 1296, 8772, 61776, 446964, 3297456, … | A001555 |
| 1n + 2n + 3n + ⋯ + 9n | 9, 45, 285, 2025, 15333, 120825, 978405, 8080425, … | A001556 |
| 1n + 2n + 3n + ⋯ + 10n | 10, 55, 385, 3025, 25333, 220825, 1978405, … | A001557 |
自然数の自然数乗 (kk) の累乗和は 1, 5, 32, 288, 3413, 50069, 873612, 17650828, …である。（A001923）
負の数を除いた 3 連続整数の 4 乗和は 17, 98, 353, 962, 2177, 4322, 7793, 13058, … である。（A160827）
同じ数の累乗和（整数乗）  \sum _{k=0}^{n}a^{k}=a^{0}+a^{1}+a^{2}+\dotsb +a^{n}={\dfrac {a^{n+1}-1}{a-1}}
| a | 数 | OEIS |
| --- | --- | --- |
| 2 | 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, … | A000225 |
| 3 | 1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524, … | A003462 |
| 4 | 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, 349525, 1398101, … | A002450 |
| 5 | 1, 6, 31, 156, 781, 3906, 19531, 97656, 488281, 2441406, … | A003463 |
| 6 | 1, 7, 43, 259, 1555, 9331, 55987, 335923, 2015539, 12093235, … | A003464 |
| 7 | 1, 8, 57, 400, 2801, 19608, 137257, 960800, 6725601, … | A023000 |
| 8 | 1, 9, 73, 585, 4681, 37449, 299593, 2396745, 19173961, … | A023001 |
| 9 | 1, 10, 91, 820, 7381, 66430, 597871, 5380840, 48427561, … | A002452 |
| 10 | 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, … | A002275 |
| 11 | 1, 12, 133, 1464, 16105, 177156, 1948717, 21435888, 235794769, … | A016123 |
| 12 | 1, 13, 157, 1885, 22621, 271453, 3257437, 39089245, 469070941, … | A016125 |
| 13 | 1, 14, 183, 2380, 30941, 402234, 5229043, 67977560, 883708281, … | A091030 |
| 14 | 1, 15, 211, 2955, 41371, 579195, 8108731, 113522235, 1589311291, … | A135519 |
| 15 | 1, 16, 241, 3616, 54241, 813616, 12204241, 183063616, 2745954241, … | A135518 |
| 16 | 1, 17, 273, 4369, 69905, 1118481, 17895697, 286331153, 4581298449, … | A131865 |
| 17 | 1, 18, 307, 5220, 88741, 1508598, 25646167, 435984840, 7411742281, … | A091045 |
| 18 | 1, 19, 343, 6175, 111151, 2000719, 36012943, 648232975, 11668193551, … | A218721 |
| 19 | 1, 20, 381, 7240, 137561, 2613660, 49659541, 943531280, 17927094321, … | A218722 |
| 20 | 1, 21, 421, 8421, 168421, 3368421, 67368421, 1347368421, … | A064108 |
| 21 | 1, 22, 463, 9724, 204205, 4288306, 90054427, 1891142968, … | A218724 |
| 22 | 1, 23, 507, 11155, 245411, 5399043, 118778947, 2613136835, … | A218725 |
| 23 | 1, 24, 553, 12720, 292561, 6728904, 154764793, 3559590240, … | A218726 |
| 24 | 1, 25, 601, 14425, 346201, 8308825, 199411801, 4785883225, … | A218727 |
| 25 | 1, 26, 651, 16276, 406901, 10172526, 254313151, 6357828776, … | A218728 |
| 26 | 1, 27, 703, 18279, 475255, 12356631, 321272407, 8353082583, … | A218729 |
| 27 | 1, 28, 757, 20440, 551881, 14900788, 402321277, 10862674480, … | A218730 |
| 28 | 1, 29, 813, 22765, 637421, 17847789, 499738093, 13992666605, … | A218731 |
| 29 | 1, 30, 871, 25260, 732541, 21243690, 616067011, 17865943320, … | A218732 |
| 30 | 1, 31, 931, 27931, 837931, 25137931, 754137931, 22624137931, … | A218733 |
同じ数の累乗和（自然数乗）  \sum _{k=1}^{n}a^{k}=a^{1}+a^{2}+\dotsb +a^{n}={\dfrac {a^{n+1}-a}{a-1}}={\dfrac {a(a^{n}-1)}{a-1}}
| a | 数 | OEIS |
| --- | --- | --- |
| 2 | 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, … | A000918 |
| 3 | 3, 12, 39, 120, 363, 1092, 3279, 9840, 29523, … | A029858 |
| 4 | 4, 20, 84, 340, 1364, 5460, 21844, 87380, 349524, 1398100, … | A080674 |
| 5 | 5, 30, 155, 780, 3905, 19530, 97655, 488280, 2441405, … | A104891 |
| 6 | 6, 42, 258, 1554, 9330, 55986, 335922, 2015538, 12093234, … | A105281 |
| 7 | 7, 56, 399, 2800, 19607, 137256, 960799, 6725600, … | A104896 |
| 8 | 8, 72, 584, 4680, 37448, 299592, 2396744, 19173960, … | A052379 |
| 9 | 9, 90, 819, 7380, 66429, 597870, 5380839, 48427560, … | A052386 |
| 10 | 10, 110, 1110, 11110, 111110, 1111110, 11111110, 111111110, … | A105279 |
| 11 | 11, 132, 1463, 16104, 177155, 1948716, 21435887, 235794768, … | A105280 |
| 12 | 12, 156, 1884, 22620, 271452, 3257436, 39089244, 469070940, … |  |