For faster navigation, this Iframe is preloading the Wikiwand page for
真因子和数列.
此条目需要扩充。 (2013年2月14日)请协助
改善这篇条目,更进一步的信息可能会在
讨论页或
扩充请求中找到。请在扩充条目后将此模板移除。
选择一个正整数
作为一个数列的开首,数列的之后的项都是上一项的真因子之和(因数函数
),即:
![{\displaystyle s_{0}=k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/929727d22b50bd7a1349b747befec57cba086707)
![{\displaystyle s_{n}=\sigma _{1}(s_{n-1})-s_{n-1))](https://wikimedia.org/api/rest_v1/media/math/render/svg/10870581d3a1aaa1c25d482a35aeb7e9cf41f24c)
这样组成的数列称为真因子和数列(aliquot sequence)。
例如取10为首项,之后是
(任何质数的唯一真因子都是1,1没有真因子)。
真因子和数列有几种可能的发展方式:
- 在1结束:好像上面的10、任何质数、18(
) ……(OEIS:A080907)
- 循环不断:对于完全数、相亲数、相亲数链的成员,真因子和数列是循环的。如果有些数本身并不属于上述提到那类数,却因为数项中有些项的真因子之和属于那类数,而有循环的真因子和数列,它们称为aspiring numbers(OEIS:A063769)。譬如:
- 完美数
:28, 28, 28...
- 四环相亲数链的成员1264460:1264460, 1547860, 1727636, 1305184, 1264460, ....
- aspiring number 95:95, 25, 6, 6, 6,
- 不循环地一直延续下去:19世纪数学家欧仁·查理·卡塔兰猜想任何真因子和数列都是按上面两种方式延续下去,但人们不但未能证明或推翻这个猜想,而且不能确定一些整数的真因子和数列。在1至1000之间,便有五个这样的数,它们称为Lehmer Five —— 276, 552, 564, 660, 966。截止2007年7月,1至105间有909个这样的数,1至106间有9466个。
{{bottomLinkPreText}}
{{bottomLinkText}}
This page is based on a Wikipedia article written by
contributors (read/edit).
Text is available under the
CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.
{{current.index+1}} of {{items.length}}
Thanks for reporting this video!
This browser is not supported by Wikiwand :(
Wikiwand requires a browser with modern capabilities in order to provide you with the best reading experience.
Please download and use one of the following browsers:
An extension you use may be preventing Wikiwand articles from loading properly.
If you're using HTTPS Everywhere or you're unable to access any article on Wikiwand, please consider switching to HTTPS (https://www.wikiwand.com).
An extension you use may be preventing Wikiwand articles from loading properly.
If you are using an Ad-Blocker, it might have mistakenly blocked our content.
You will need to temporarily disable your Ad-blocker to view this page.
✕
This article was just edited, click to reload
Please click Add in the dialog above
Please click Allow in the top-left corner,
then click Install Now in the dialog
Please click Open in the download dialog,
then click Install
Please click the "Downloads" icon in the Safari toolbar, open the first download in the list,
then click Install
{{::$root.activation.text}}
Follow Us
Don't forget to rate us
Oh no, there's been an error
Please help us solve this error by emailing us at
support@wikiwand.com
Let us know what you've done that caused this error, what browser you're using, and whether you have any special extensions/add-ons installed.
Thank you!