水仙花数

数论中,水仙花数Narcissistic number[1][2],也被稱為超完全数字不变数pluperfect digital invariant, PPDI[3]自戀數自幂數阿姆斯壯數阿姆斯特朗數Armstrong number[4] ,用来描述一个N位非负整数,其各位数字的N次方和等于该数本身。

水仙花数的定义

设有自然数n,d为该自然数各位数字,即 n = dkdk-1...d1 ,则有:

n = dk·10k-1 + dk-1·10k-2 + ... + d2·10 + d1,

如果该自然数n满足条件:

n = dkk + dk-1k + ... + d2k + d1k.

则这个自然数就被称为超完全数字不变数。 例如153、370、371及407就是三位超完全数字不变数,其各个数之立方和等于该数:

153 = 13 + 53 + 33
370 = 33 + 73 + 03
371 = 33 + 73 + 13
407 = 43 + 03 + 73

若將條件放寬,一個N位数,其各个数之M次方和等于该数,M和N不一定相等,這樣的數稱為完全數字不變數(perfect digital invariant)[5][2],例如數字4150等於各位數字的5次方。

4150 = 45 + 15 + 55 + 05,

水仙花数一定是完全數字不變數,但完全數字不變數不一定是水仙花数。 严格意义来说水仙花数指三位数。

部分水仙花数

所有数字均以b为底。

水仙花数 # 循环 OEIS sequence(s)
20,12
30, 1, 2, 12, 22, 1226
40, 1, 2, 3, 130, 131, 203, 223, 313, 332, 1103, 330312A010344 and A010343
50, 1, 2, 3, 4, 23, 33, 103, 433, 2124, 2403, 3134, 124030, 124031, 242423, ...18

1234 → 2404 → 4103 → 2323 → 1234

3424 → 4414 → 11034 → 20034 → 20144 → 31311 → 3424

1044302 → 2110314 → 1044302

1043300 → 1131014 → 1043300

A010346
60, 1, 2, 3, 4, 5, 243, 514, 14340, 14341, 14432, 23520, 23521, 44405, 435152, 5435254, 12222215, 555435035 ...31

44 → 52 → 45 → 105 → 330 → 130 → 44

13345 → 33244 → 15514 → 53404 → 41024 → 13345

14523 → 32253 → 25003 → 23424 → 14523

2245352 → 3431045 → 2245352

12444435 → 22045351 → 30145020 → 13531231 → 12444435

115531430 → 230104215 → 115531430

225435342 → 235501040 → 225435342

A010348
70, 1, 2, 3, 4, 5, 6, 13, 34, 44, 63, 250, 251, 305, 505, 12205, 12252, 13350, 13351, 15124, 36034, ...60A010350
80, 1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661, ...63A010354 and A010351
90, 1, 2, 3, 4, 5, 6, 7, 8, 45, 55, 150, 151, 570, 571, 2446, 12036, 12336, 14462, ...59A010353
100, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, ...89A005188
110, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, 56, 66, 105, 307, 708, 966, A06, A64, 8009, 11720, 11721, 12470, ...135A0161948
120, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 25, A5, 577, 668, A83, 14765, 938A4, 369862, A2394A, ...88A161949
130, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, 14, 36, 67, 77, A6, C4, 490, 491, 509, B85, 3964, 22593, 5B350, ...202A0161950
140, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, 136, 409, 74AB5, 153A632, ...103A0161951
150, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, 78, 88, C3A, D87, 1774, E819, E829, 7995C, 829BB, A36BC, ...203A0161952
160, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 156, 173, 208, 248, 285, 4A5, 5B0, 5B1, 60B, 64B, 8C0, 8C1, 99A, AA9, AC3, CA8, E69, EA0, EA1, ...294A161953

十進制下的水仙花数

十进制的水仙花數共有89個,最大的是

115,132,219,018,763,992,565,095,597,973,971,522,401

共有39位數。[6]

完整的十进制水仙花数列表如下:OEIS中的数列A005188

  1. 0
  2. 1
  3. 2
  4. 3
  5. 4
  6. 5
  7. 6
  8. 7
  9. 8
  10. 9
  11. 153
  12. 370
  13. 371
  14. 407
  15. 1634
  16. 8208
  17. 9474
  18. 54748
  19. 92727
  20. 93084
  21. 548834
  22. 1741725
  23. 4210818
  24. 9800817
  25. 9926315
  26. 24678050
  27. 24678051
  28. 88593477
  29. 146511208
  30. 472335975
  31. 534494836
  32. 912985153
  33. 4679307774
  34. 32164049650
  35. 32164049651
  36. 40028394225
  37. 42678290603
  38. 44708635679
  39. 49388550606
  40. 82693916578
  41. 94204591914
  42. 28116440335967
  43. 4338281769391370
  44. 4338281769391371
  45. 21897142587612075
  46. 35641594208964132
  47. 35875699062250035
  48. 1517841543307505039
  49. 3289582984443187032
  50. 4498128791164624869
  51. 4929273885928088826
  52. 63105425988599693916
  53. 128468643043731391252
  54. 449177399146038697307
  55. 21887696841122916288858
  56. 27879694893054074471405
  57. 27907865009977052567814
  58. 28361281321319229463398
  59. 35452590104031691935943
  60. 174088005938065293023722
  61. 188451485447897896036875
  62. 239313664430041569350093
  63. 1550475334214501539088894
  64. 1553242162893771850669378
  65. 3706907995955475988644380
  66. 3706907995955475988644381
  67. 4422095118095899619457938
  68. 121204998563613372405438066
  69. 121270696006801314328439376
  70. 128851796696487777842012787
  71. 174650464499531377631639254
  72. 177265453171792792366489765
  73. 14607640612971980372614873089
  74. 19008174136254279995012734740
  75. 19008174136254279995012734741
  76. 23866716435523975980390369295
  77. 1145037275765491025924292050346
  78. 1927890457142960697580636236639
  79. 2309092682616190307509695338915
  80. 17333509997782249308725103962772
  81. 186709961001538790100634132976990
  82. 186709961001538790100634132976991
  83. 1122763285329372541592822900204593
  84. 12639369517103790328947807201478392
  85. 12679937780272278566303885594196922
  86. 1219167219625434121569735803609966019
  87. 12815792078366059955099770545296129367
  88. 115132219018763992565095597973971522400
  89. 115132219018763992565095597973971522401

十二進制下的水仙花數

OEIS中的数列A161949

25:22+52=25
A5:A2+52=A5
577:53+73+73=577
668:63+63+83=668
A83:A3+83+33=A83
14765:15+45+75+65+55=14765
938A4:95+35+85+A5+45=938A4

顯然,任何一位數(從1到B)都是水仙花數,另外,在十二進制中,不存在四位數的水仙花數。

参考资料

  1. 埃里克·韦斯坦因. . MathWorld.
  2. Perfect and PluPerfect Digital Invariants 存檔,存档日期2007-10-10. by Scott Moore
  3. PPDI (Armstrong) Numbers by Harvey Heinz
  4. Armstrong Numbersl by Dik T. Winter
  5. PDIs by Harvey Heinz
  6. Weisstein, Eric W. . mathworld.wolfram.com. [2019-06-10]. (原始内容存档于2018-01-20) (英语).
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