有关R(n)的一个猜想(英文)

On a conjecture concerning R( n)

针对一个关于算数函数R(n)的有趣的猜想。R(n)是一个与所有可以整除n的正整数之和有关的函数。首先利用唯一分解定理建立一些有关R(n)单调性的预备性结果。通过对n做唯一分解,对某类特殊的n,得到一些R(n)的上下界估计。这样,在某种意义上,证明了猜想。其次得到了对于某类n的R(n)的上无界性。给出了R(n)=1的充要条件。事实上,R(n)=1当且仅当n为素数。其次,给出对于某些n,使得R(n)=2的充要条件。利用预备知识,进一步研究了R(n)的单调性。得出对于固定的k≥2,至多有一个这样的n使得R(n)=k这样的结论。最后给出使得R(n)=2的具体的n的例子,并计算了10 000以内的R(n)的数值,这样在10 000以内,验证了猜想。

An interesting conjecture about the properties of an arithmetic function R （n）, relevant to the sum of all the divisors of n, is dealt with. First, some preliminary results on the which is monotonicity of R （n） are established by using the unique factorization of n. For some specific types of n in terms of the unique factorization of n, some upper and lower bounds of R（n） are obtained. In this way, the conjecture is proven in some sense. Second, the unboundedness of R（n） for some other types of n is mainly derived, and the sufficient and necessary condition for R （n） = 1 is given. In fact, R （n） = 1 if and only if n is a prime number. Besides, for some n, some sufficient and necessary conditions for R （n） = 2 are given. Using the preliminaries, the monotonicity properties of R （n） are further investigated. It is shown that there is at most one n such that R（n） = k for some k≥2. Finally, some examples of n such that R（n） =2 is given, and R（n） is offered when n≤10 000, which verifies the conjecture when n≤10000.