循环单位是由数字 1 重复组成的数字。“循环单位”一词由 Beiler (1966) 创造，他也是已知因子的第一个制表者。
在十进制中，循环单位具有以下形式
因此，循环单位 
恰好有 
位十进制数字。 令人惊讶的是，循环单位的平方 
给出了 Demlo 数， 
, 
, 
, ... (OEIS A002275 和 A002477)。
对于 
, 2, ... 的十进制循环单位，因子的数量为 1, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, ... (OEIS A046053)。
是素数的循环单位被称为循环单位素数。
循环单位可以推广到基数 
，从而得到形如 of the form
 |
(3)
|
下表总结了这些特殊情况。
循环单位的概念也可以扩展到负基数。除了要求 
为奇数外，其数学原理非常相似 (Dubner and Granlund 2000)。
 | OEIS |  -循环单位 |
 | A066443 | 1, 7, 61, 547, 4921, 44287, 398581, ... |
 | A007583 | 1, 3, 11, 43, 171, 683, 2731, ... |
| 2 | A000225 | 1,
3, 7, 15, 31, 63, 127, ... |
| 3 | A003462 | 1, 4, 13, 40, 121,
364, ... |
| 4 | A002450 | 1, 5, 21, 85, 341, 1365, ... |
| 5 | A003463 | 1,
6, 31, 156, 781, 3906, ... |
| 6 | A003464 | 1, 7, 43, 259, 1555,
9331, ... |
| 7 | A023000 | 1, 8, 57, 400, 2801, 19608, ... |
| 8 | A023001 | 1,
9, 73, 585, 4681, 37449, ... |
| 9 | A002452 | 1, 10, 91, 820, 7381,
66430, ... |
| 10 | A002275 | 1, 11, 111, 1111, 11111, ... |
| 11 | A016123 | 1,
12, 133, 1464, 16105, 177156, ... |
| 12 | A016125 | 1,
13, 157, 1885, 22621, 271453, ... |
另请参阅
Cunningham Number,
Demlo Number,
Fermat Number,
Mersenne Number,
Repdigit,
Repunit Prime,
Smith Number
使用 Wolfram|Alpha 探索
参考文献
Beiler, A. H. "11111...111." Ch. 11 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966.Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of b-n+/-1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc., 1988.Dudeney, H. E. The Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson and Sons, 1949.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 85-86, 1984.Granlund, T. "Repunits." http://www.swox.com/gmp/repunit.html.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 152-153, 1979.Ribenboim, P. "Repunits and Similar Numbers." §5.5 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 350-354, 1996.Sloane, N. J. A. Sequences A000043/M0672, A000225/M2655, A000978, A001562, A002275, A002477/M5386, A002450/M3914, A002452/M4733, A003462/M3463, A007583, A007658, A003463/M4209, A003464/M4425, A004023/M2114, A004061/M2620, A004062/M0861, A004063/M3836, A004064/M0744, A005808/M5032, A016123, A016125, A023000, A023001, A028491/M2643, A046053, A057171, A057172, A057173, A057175, A057177, A057178, A066443, and A084740 in "The On-Line Encyclopedia of Integer Sequences."Yates, S. "Peculiar Properties of Repunits." J. Recr. Math. 2, 139-146, 1969.Yates, S. "The Mystique of Repunits." Math. Mag. 51, 22-28, 1978.Yates, S. Repunits and Reptends. Delray Beach, FL: S. Yates, 1982.在 Wolfram|Alpha 中引用
循环单位
请引用本文为
Weisstein, Eric W. "Repunit." 来自 MathWorld——Wolfram Web 资源。 https://mathworld.net.cn/Repunit.html
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