关于奇素数方幂中的孤立数

设n是正整数,用σ(n)表示n的所有正因数的和。对于给定的正整数a,如果不存在正整数b适合σ(a)=σ(b)=a+b,则称a是孤立数。文章运用初等数论的方法证明了pr都是孤立数。这里p为奇素数,满足p>2r~(1+ε),0<ε≤1,ε是任意实数,r是正整数,满足r>((1+ε)/ε)~1/ε

On the Anti-sociable Numbers in Powers of Odd Primes

Let n be a positive integer,the sum of all the positive divisors of n is defined by σ(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:σ(a)=σ(b)=a+b.By using some elementary number theory methods in this paper,the fact that with p>2r~(1+ε),0<ε≤1,if p is an odd prime,ε is a real number,and r is a positive integer with r>((1+ε)/ε)~1/ε,is proved.