Prove that there exist infinitely many odd abundant numbers.
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?