For any integer n show that there exist integers k and j such that φ(k)+σ(j)=n

Printable View

- May 7th 2010, 08:42 AMogajawasunguFor any integer n
For any integer n show that there exist integers k and j such that φ(k)+σ(j)=n

- May 10th 2010, 08:30 AMSwlabr
In words, show that for all n there exists integers k and j such that the number of divisors of j plus the number of numbers coprime to and less than k is n.

So, I would note that so taking , .

However, this is only for the natural numbers and zero...not the whole integers. You can't get a negative number of divisors!

It is also a somewhat boring result - why the Euler totient function if we can just forget about it?...

Have you perhaps missed out something from the question? - May 10th 2010, 09:01 AMogajawasungu
Sorry, the question should read; find all integers n such that σ(n)+φ(n)=2n

- May 10th 2010, 11:06 AMchiph588@
- May 10th 2010, 06:57 PMhollywood
If p is prime, then sigma(p)=p+1 (since there are only the two factors, 1 and p). Also, phi(p)=p-1 since all of the integers from 1 to p-1 are coprime to p, so sigma(p)+phi(p)=2p.

I think it's probably true, but I can't see a way to prove that there are no composite n with sigma(n)+phi(n)=2n.

- Hollywood - May 10th 2010, 09:07 PMBacteriusQuote:

Originally Posted by**hollywood**

So

Oops, I just wiped out all semiprimes :(

I believe there is a general proof that can be built around this with infinitely many prime factors instead of a semiprime. I've seen it but unfortunately forgot it, I'll have to check it out some time.

**Conclusion**: only prime numbers satisfy this statement, along with 1, assuming the validity and existence of the GFP (Generalized Forgotten Proof :D) - May 10th 2010, 10:25 PMundefined