形如2^rp1p2…p_k（k≥2）的孤立数

Anti-sociable Number With the Form 2^r p1p2…pk（k≥2）

对于正整数n,设δ(n)是n的不同约数之和.对于给定的正整数a,如果不存在正整数b满足δ(a)=δ(b)=a+b,则称a是孤立数.由于相亲数和孤立数与完全数等著名数学难题有着直接的联系,所以它们一直是数论中引人关注的研究课题.利用素数个数的估计的相关结论证明了对于任意给定正整数r或者k,均存在无穷多个形如的孤立数,其中p1,p2,…,pk为相异的素数.

For any positive integer n, letδ（n） to be the sum of all the different divisors of n. Suppose that a is a positive integer,then a is called an Anti- sociable number if only there is not the positive integer b：δ（a）=δ（b）=a＋b. Because Amicable pair and Anti- sociable number with the Perfect number of famous mathematical problems has a direct connection, so they have been interesting research topic in number theory. Based on the estimate of prime number of related conclusions, we prove that for any positive integer r or k, there are infinity many anti- sociable number with the from 2r p1p2…pk（k≥2）,wherepi（i= 1,2…k）is the distinct primes.