Prove that there exist infinitely many odd abundant numbers.

Proof.

Consider the integers n = 945k, where k is any positive integers not divisible by 2,3,5, or 7.

Since 945 = (3^3)(5)(7), so gcd(945,k)=1, implies that

Therefore I have

Now, I claim that , and I'm using induction to prove it.

Certainly, , so the claim is true when k=1.

Suppose the claim is true for k=t, we then have

Now, I want to show that , how would I do that?

Thanks.