盧卡斯-卡米切爾數
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在 數學中, 盧卡斯-卡米切爾數 是一個正合數 n 滿足,如果 p 是 n 的質因子, 那麼 p + 1 是n + 1的因子. 它以 愛德華·盧卡斯 和 羅伯特·丹尼·卡邁克爾命名.
按照約定, 一個數被稱作盧卡斯-卡米切爾數若且唯若它是奇數並且是 無平方數因數的數 (不能被一個質數的平方整除), 否則任何質數的立方,像8和27, 都將成為盧卡斯-卡米切爾數 (因為 n3 + 1 = (n + 1)(n2 − n + 1) 一定可以被 n + 1整除).
滿足條件的最小的數是 399 = 3 × 7 × 19; 399+1 = 400; 3+1, 7+1 和 19+1 都是400的因子.
盧卡斯-卡米切爾數的前幾個數及他們的因子是 (OEIS數列A006972):
399 | = 3 × 7 × 19 |
935 | = 5 × 11 × 17 |
2015 | = 5 × 13 × 31 |
2915 | = 5 × 11 × 53 |
4991 | = 7 × 23 × 31 |
5719 | = 7 × 19 × 43 |
7055 | = 5 × 17 × 83 |
8855 | = 5 × 7 × 11 × 23 |
12719 | = 7 × 23 × 79 |
18095 | = 5 × 7 × 11 × 47 |
20705 | = 5 × 41 × 101 |
20999 | = 11 × 23 × 83 |
22847 | = 11 × 31 × 67 |
29315 | = 5 × 11 × 13 × 41 |
31535 | = 5 × 7 × 17 × 53 |
46079 | = 11 × 59 × 71 |
51359 | = 7 × 11 × 23 × 291 |
60059 | = 19 × 29 × 109 |
63503 | = 11 × 23 × 251 |
67199 | = 11 × 41 × 149 |
73535 | = 5 × 7 × 11 × 191 |
76751 | = 23 × 47 × 71 |
80189 | = 17 × 53 × 89 |
81719 | = 11 × 17 × 19 × 23 |
88559 | = 19 × 59 × 79 |
90287 | = 17 × 47 × 113 |
104663 | = 13 × 83 × 97 |
117215 | = 5 × 7 × 17 × 197 |
120581 | = 17 × 41 × 173 |
147455 | = 5 × 7 × 11 × 383 |
152279 | = 29 × 59 × 89 |
155819 | = 19 × 59 × 139 |
162687 | = 3 × 7 × 61 × 127 |
191807 | = 7 × 11 × 47 × 53 |
194327 | = 7 × 17 × 23 × 71 |
196559 | = 11 × 107 × 167 |
214199 | = 23 × 67 × 139 |
218735 | = 5 × 11 × 41 × 97 |
230159 | = 47 × 59 × 83 |
265895 | = 5 × 7 × 71 × 107 |
357599 | = 11 × 19 × 29 × 59 |
388079 | = 23 × 47 × 359 |
390335 | = 5 × 11 × 47 × 151 |
482143 | = 31 × 103 × 151 |
588455 | = 5 × 7 × 17 × 23 × 43 |
653939 | = 11 × 13 × 17 × 269 |
663679 | = 31 × 79 × 271 |
676799 | = 19 × 179 × 199 |
709019 | = 17 × 179 × 233 |
741311 | = 53 × 71 × 197 |
760655 | = 5 × 7 × 103 × 211 |
761039 | = 17 × 89 × 503 |
776567 | = 11 × 227 × 311 |
798215 | = 5 × 11 × 23 × 631 |
880319 | = 11 × 191 × 419 |
895679 | = 17 × 19 × 47 × 59 |
913031 | = 7 × 23 × 53 × 107 |
966239 | = 31 × 71 × 439 |
966779 | = 11 × 179 × 491 |
973559 | = 29 × 59 × 569 |
1010735 | = 5 × 11 × 17 × 23 × 47 |
1017359 | = 7 × 23 × 71 × 89 |
1097459 | = 11 × 19 × 59 × 89 |
1162349 | = 29 × 149 × 269 |
1241099 | = 19 × 83 × 787 |
1256759 | = 7 × 17 × 59 × 179 |
1525499 | = 53 × 107 × 269 |
1554119 | = 7 × 53 × 59 × 71 |
1584599 | = 37 × 113 × 379 |
1587599 | = 13 × 97 × 1259 |
1659119 | = 7 × 11 × 29 × 743 |
1707839 | = 7 × 29 × 47 × 179 |
1710863 | = 7 × 11 × 17 × 1307 |
1719119 | = 47 × 79 × 463 |
1811687 | = 23 × 227 × 347 |
1901735 | = 5 × 11 × 71 × 487 |
1915199 | = 11 × 13 × 59 × 227 |
1965599 | = 79 × 139 × 179 |
2048255 | = 5 × 11 × 167 × 223 |
2055095 | = 5 × 7 × 71 × 827 |
2150819 | = 11 × 19 × 41 × 251 |
2193119 | = 17 × 23 × 71 × 79 |
2249999 | = 19 × 79 × 1499 |
2276351 | = 7 × 11 × 17 × 37 × 47 |
2416679 | = 23 × 179 × 587 |
2581319 | = 13 × 29 × 41 × 167 |
2647679 | = 31 × 223 × 383 |
2756159 | = 7 × 17 × 19 × 23 × 53 |
2924099 | = 29 × 59 × 1709 |
3106799 | = 29 × 149 × 719 |
3228119 | = 19 × 23 × 83 × 89 |
3235967 | = 7 × 17 × 71 × 383 |
3332999 | = 19 × 23 × 29 × 263 |
3354695 | = 5 × 17 × 61 × 647 |
3419999 | = 11 × 29 × 71 × 151 |
3441239 | = 109 × 131 × 241 |
3479111 | = 83 × 167 × 251 |
3483479 | = 19 × 139 × 1319 |
3700619 | = 13 × 41 × 53 × 131 |
3704399 | = 47 × 269 × 293 |
3741479 | = 7 × 17 × 23 × 1367 |
4107455 | = 5 × 11 × 17 × 23 × 191 |
4285439 | = 89 × 179 × 269 |
4452839 | = 37 × 151 × 797 |
4587839 | = 53 × 107 × 809 |
4681247 | = 47 × 103 × 967 |
4853759 | = 19 × 23 × 29 × 383 |
4874639 | = 7 × 11 × 29 × 37 × 59 |
5058719 | = 59 × 179 × 479 |
5455799 | = 29 × 419 × 449 |
5669279 | = 7 × 11 × 17 × 61 × 71 |
5807759 | = 83 × 167 × 419 |
6023039 | = 11 × 29 × 79 × 239 |
6514199 | = 43 × 197 × 769 |
6539819 | = 11 × 13 × 19 × 29 × 83 |
6656399 | = 29 × 89 × 2579 |
6730559 | = 11 × 23 × 37 × 719 |
6959699 | = 59 × 179 × 659 |
6994259 | = 17 × 467 × 881 |
7110179 | = 37 × 41 × 43 × 109 |
7127999 | = 23 × 479 × 647 |
7234163 | = 17 × 41 × 97 × 107 |
7274249 | = 17 × 449 × 953 |
7366463 | = 13 × 23 × 71 × 347 |
8159759 | = 19 × 29 × 59 × 251 |
8164079 | = 7 × 11 × 229 × 463 |
8421335 | = 5 × 13 × 23 × 43 × 131 |
8699459 | = 43 × 307 × 659 |
8734109 | = 37 × 113 × 2089 |
9224279 | = 53 × 269 × 647 |
9349919 | = 19 × 29 × 71 × 239 |
9486399 | = 3 × 13 × 79 × 3079 |
9572639 | = 29 × 41 × 83 × 97 |
9694079 | = 47 × 239 × 863 |
9868715 | = 5 × 43 × 197 × 233 |
最小的有5個因子的盧卡斯-卡米切爾數是 588455 = 5 × 7 × 17 × 23 × 43.
參考
- PlanetMath (頁面存檔備份,存於網際網路檔案館)