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# Number Theory watch

1. Stuck on this question, prove or disprove sigma(n) is less than or equal to n^2 for n>0 where sigma(n) is the sum of the divisiors of n, so sigma(12)=1+2+3+4+6+12

I can tell by intuition this is true, but don't know how to prove it
2. (Original post by Jimmath)
Stuck on this question, prove or disprove sigma(n) is less than or equal to n^2 for n>0 where sigma(n) is the sum of the divisiors of n, so sigma(12)=1+2+3+4+6+12

I can tell by intuition this is true, but don't know how to prove it
It might be helpful to consider the sum of the reciprocals of the divisors.

Edit: Or better, consider what is the maximum number of divisors and what is the biggest they can be!
3. (Original post by Jimmath)
Stuck on this question, prove or disprove sigma(n) is less than or equal to n^2 for n>0 where sigma(n) is the sum of the divisiors of n, so sigma(12)=1+2+3+4+6+12

I can tell by intuition this is true, but don't know how to prove it
As an extension, have this problem:

Let be a positive integer with divisors .
Prove that

As a further extension, determine when is a divisor of
4. (Original post by Renzhi10122)
As an extension, have this problem:

Let be a positive integer with divisors .
Prove that
Consider in relation to n,

What is ?
What is the largest value can be?

Continue.

edit: I assume you are saying you have that problem, rather than proposing it for the OP. If not, ignore this.
5. (Original post by ghostwalker)
Consider in relation to n,

What is ?
What is the largest value can be?

Continue.

edit: I assume you are saying you have that problem, rather than proposing it for the OP. If not, ignore this.
Nah, ive solved this one, OP's problem simply reminded me of this.

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