水仙花数
在数论中,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、阿姆斯壯數或阿姆斯特朗數(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) | |
---|---|---|---|---|
2 | 0,1 | 2 | ||
3 | 0, 1, 2, 12, 22, 122 | 6 | ||
4 | 0, 1, 2, 3, 130, 131, 203, 223, 313, 332, 1103, 3303 | 12 | A010344 and A010343 | |
5 | 0, 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 |
6 | 0, 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 |
7 | 0, 1, 2, 3, 4, 5, 6, 13, 34, 44, 63, 250, 251, 305, 505, 12205, 12252, 13350, 13351, 15124, 36034, ... | 60 | A010350 | |
8 | 0, 1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661, ... | 63 | A010354 and A010351 | |
9 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 45, 55, 150, 151, 570, 571, 2446, 12036, 12336, 14462, ... | 59 | A010353 | |
10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, ... | 89 | A005188 | |
11 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, 56, 66, 105, 307, 708, 966, A06, A64, 8009, 11720, 11721, 12470, ... | 135 | A0161948 | |
12 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 25, A5, 577, 668, A83, 14765, 938A4, 369862, A2394A, ... | 88 | A161949 | |
13 | 0, 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, ... | 202 | A0161950 | |
14 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, 136, 409, 74AB5, 153A632, ... | 103 | A0161951 | |
15 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, 78, 88, C3A, D87, 1774, E819, E829, 7995C, 829BB, A36BC, ... | 203 | A0161952 | |
16 | 0, 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, ... | 294 | A161953 |
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)
- 0
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- 8
- 9
- 153
- 370
- 371
- 407
- 1634
- 8208
- 9474
- 54748
- 92727
- 93084
- 548834
- 1741725
- 4210818
- 9800817
- 9926315
- 24678050
- 24678051
- 88593477
- 146511208
- 472335975
- 534494836
- 912985153
- 4679307774
- 32164049650
- 32164049651
- 40028394225
- 42678290603
- 44708635679
- 49388550606
- 82693916578
- 94204591914
- 28116440335967
- 4338281769391370
- 4338281769391371
- 21897142587612075
- 35641594208964132
- 35875699062250035
- 1517841543307505039
- 3289582984443187032
- 4498128791164624869
- 4929273885928088826
- 63105425988599693916
- 128468643043731391252
- 449177399146038697307
- 21887696841122916288858
- 27879694893054074471405
- 27907865009977052567814
- 28361281321319229463398
- 35452590104031691935943
- 174088005938065293023722
- 188451485447897896036875
- 239313664430041569350093
- 1550475334214501539088894
- 1553242162893771850669378
- 3706907995955475988644380
- 3706907995955475988644381
- 4422095118095899619457938
- 121204998563613372405438066
- 121270696006801314328439376
- 128851796696487777842012787
- 174650464499531377631639254
- 177265453171792792366489765
- 14607640612971980372614873089
- 19008174136254279995012734740
- 19008174136254279995012734741
- 23866716435523975980390369295
- 1145037275765491025924292050346
- 1927890457142960697580636236639
- 2309092682616190307509695338915
- 17333509997782249308725103962772
- 186709961001538790100634132976990
- 186709961001538790100634132976991
- 1122763285329372541592822900204593
- 12639369517103790328947807201478392
- 12679937780272278566303885594196922
- 1219167219625434121569735803609966019
- 12815792078366059955099770545296129367
- 115132219018763992565095597973971522400
- 115132219018763992565095597973971522401
参考资料
- 埃里克·韦斯坦因. . MathWorld.
- Perfect and PluPerfect Digital Invariants 的存檔,存档日期2007-10-10. by Scott Moore
- PPDI (Armstrong) Numbers by Harvey Heinz
- Armstrong Numbersl by Dik T. Winter
- PDIs by Harvey Heinz
- Weisstein, Eric W. . mathworld.wolfram.com. [2019-06-10]. (原始内容存档于2018-01-20) (英语).
- Rose, Colin (2005), Radical narcissistic numbers, Journal of Recreational Mathematics, 33(4), 2004-2005, pages 250-254.
- Perfect Digital Invariants by Walter Schneider
- On a curious property of 3435 页面存档备份,存于 by Daan van Berkel