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1.
Let k ≥ 2 be an integer, and let σ(n) denote the sum of the positive divisors of an integer n. We call n a quasi-multiperfect number if σ(n) = kn + 1. In this paper, we give some necessary properties of them.  相似文献   

2.
Let k ≥ 2 be an integer, and let σ(n) denote the sum of the positive divisors of an integer n. We call n a quasi-multiperfect number if σ(n) = kn + 1. In this paper, we give some necessary properties of quasi-multiperfect numbers with four different prime divisors.  相似文献   

3.
In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 σ 1, and each given integer b 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.  相似文献   

4.
Let k ≥ 2 be an integer, and let a(n) denote the sum of the positive divisors of an integer n. We call n a quasi-multiperfect number if a(n) = kn + 1. In this paper, we give some necessary properties of quasi-multiperfect numbers with four different prime divisors.  相似文献   

5.
The smallest degree sum that yields potentially Kr,r-graphic sequences   总被引:2,自引:0,他引:2  
We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(Kr,r, n) such that every n-term graphic sequence π = (d1, d2,..., dn) with term sum σ(π) = d1 + d2 +…+ dn ≥σ(Kr,r, n) is potentially Kr,r-graphic, where Kr,r is an r × r complete bipartite graph, i.e. πr has a realization G containing Kr,r as its subgraph. In this paper, the values σ(Kr,r,n) for even r and n ≥ 4r2 - r - 6 and for odd r and n ≥ 4r2 + 3r - 8 are determined.  相似文献   

6.
Let σ(k, n) be the smallest even integer such that each n-term positive graphic sequence with term sum at least σ(k, n) can be realized by a graph containing a clique of k + 1 vertices. Erdos et al. (Graph Theory, 1991, 439-449) conjectured that σ(k, n) = (k - 1)(2n- k) + 2. Li et al. (Science in China, 1998, 510-520) proved that the conjecture is true for k 〉 5 and n ≥ (k2) + 3, and raised the problem of determining the smallest integer N(k) such that the conjecture holds for n ≥ N(k). They also determined the values of N(k) for 2 ≤ k ≤ 7, and proved that [5k-1/2] ≤ N(k) ≤ (k2) + 3 for k ≥ 8. In this paper, we determine the exact values of σ(k, n) for n ≥ 2k+3 and k ≥ 6. Therefore, the problem of determining σ(k, n) is completely solved. In addition, we prove as a corollary that N(k) -= [5k-1/2] for k ≥6.  相似文献   

7.
Let p(n) denote the largest prime factor of an integer n≥2, and let Q(n) denote the largest prime power which divides n≥2. The purpose of this paper is to give asymptotic formula for the sum.  相似文献   

8.
Let(X, f) be a topological dynamical system, where X is a nonempty compact and metrizable space with the metric d and f : X → X is a continuous map. For any integer n ≥ 2, denote the product space by X(n)= X ×× X n times. We say a system(X, f) is generally distributionally n-chaotic if there exists a residual set D ? X(n)such that for any point x =(x1,, xn) ∈ D,lim infk→∞#({i : 0 ≤ i ≤ k- 1, min{d(fi(xj), fi(xl)) : 1 ≤ j = l ≤ n} δ0})k= 0for some real number δ0 0 and lim sup k→∞#({i : 0 ≤ i ≤ k- 1, max{d(fi(xj), fi(xl)) : 1 ≤ j = l ≤ n} δ})k= 1for any real number δ 0, where #() means the cardinality of a set. In this paper, we show that for each integer n ≥ 2, there exists a system(X, σ) which satisfies the following conditions:(1)(X, σ) is transitive;(2)(X, σ) is generally distributionally n-chaotic, but has no distributionally(n + 1)-tuples;(3) the topological entropy of(X, σ) is zero and it has an IT-tuple.  相似文献   

9.
A graph is one-regular if its automorphism group acts regularly on the set of its arcs.Let n be a square-free integer.In this paper,we show that a cubic one-regular graph of order 2n exists if and only if n=3~tp1p2…p_s≥13,where t≤1,s≥1 and p_i's are distinct primes such that 3|(P_i—1). For such an integer n,there are 2~(s-1) non-isomorphic cubic one-regular graphs of order 2n,which are all Cayley graphs on the dihedral group of order 2n.As a result,no cubic one-regular graphs of order 4 times an odd square-free integer exist.  相似文献   

10.
Our work pays much attention to the instability of an nth-order differential equation whose coefficients are not all constants, where n is an odd integer and n ≥ 3. Finally, we obtained a sufficient condition of the instability.  相似文献   

11.
对于任意正整数n,设幂p的原数函数Sp(n)定义为Sp(n)=min{m:pn|m!},其中p为素数.本文目的是运用初等方法研究函数Sp(n2)与Sp(n)之间的关系,并得到一个有趣的恒等式.  相似文献   

12.
关于数论函数σ(n)的一个注记   总被引:2,自引:0,他引:2  
对于两个不相同的正整数m和n,如果满足σ(m)=σ(n)=m n,则称之为一对亲和数,这里σ(n)=∑d|nd.本文给出了f(x,y)=x2x y2x(x>y≥1,(x,y)=1)不与任何正整数构成亲和数对的结论,这里x,y具有不同的奇偶性,即,关于z的方程σ(f,(x,y))=σ(z)=f(x,y) z不存在正整数解.  相似文献   

13.
对于两个不相同的正整数$m$和$n$, 如果满足$\sigma(m)=\sigma(n)=m+n$, 则称之为一对亲和数, 这里$\sigma(n)=\sum_{d|n}d$.本文给出了$f(x,y)=x^{2^{x}}+y^{2^{x}}(x>y\geq{1},(x,y)=1)$不与任何正整数构成亲和数对的结论, 这里$x$,$y$具有不同的奇偶性, 即, 关于$z$的方程$\sigma(f(x,y))=\sigma(z)=f(x,y)+z$不存在正整数解.  相似文献   

14.
给出数列{n/(n!)~(1/n)}极限的八种灵活新颖的求解方法,进一步讨论了它的教学价值.  相似文献   

15.
Several new and sharp inequalities involving the constant e and the sequence (1 + 1/n) n are proved.  相似文献   

16.
对于任意给定的素数p及正整数n,我们定义幂p的原数函数Sp(n)为最小的正整数m使得pn|m!.即就是:Sp(n)=min{m:pn|m!}.本文的主要目的是利用初等方法研究函数Sp(n)的算术性质,并得到一个有趣的恒等式.  相似文献   

17.
在Jeismanowicz猜想的基础上,利用初等方法证明了对任意的正整数n, Diophantine方程(44n)x+(117n)y=(125n)z 仅有正整数解(x, y, z)=(2,2,2)。  相似文献   

18.
The(d,k)-dominating number is a new measure to characterize reliability of resourcesharing in fault tolerant networks.This paper obtains that the(n,2n)-dominating number of the n-dimensional undirected toroidal mesh C(3,3,…,3)is equal to 3(n≥3).  相似文献   

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