All* completely additive functions form the same group of perfect numbers.

[Žiga Pirc]

Hello!

Yesterday I discovered that A001414 and A001222 generate the same perfect numbers i.e. n is an A001222-perfect number if 2*A001222(n) = Sum_{d|n} A001222(d). They both give A007422, numbers such that product of their divisors is their square or numbers with exactly 4 divisors along with 1.

Checking some other completely additive sequences that are listed it seems that all generate the same numbers. So, is a number a-perfect, where a is completely additive function if and only if it's 1 or has exactly 4 divisors?

Later yesterday and today I created a proof, that confirms the conjecture, but only for a(n) != 0, n>1. For cases where a(n) can be 0, the conjecture isn't true. For example, the 0 function/sequence (A000004) is completely additive, but it's perfect numbers are all the integers.

First we prove that numbers with 4 divisors are a-perfect. We have 2 cases: p*q, for diffrent primes p and q, and p^3. 2*a(p*q) = 2*(a(p)+a(q)). Sum of a(d) is 0+a(p)+a(q)+a(p*q), so it is equal to 2*a(p*q). Same for 2*a(p^3) = 6*a(p) which is equal to 0+a(p)+2*a(p)+3*a(p). 
1 is also an a-perfect number since a(1) will always be 0, 2*0 = 0.

Now to prove that a-perfect numbers above 1 have 4 divisors. We define n = p1^a1*p2^a2*...pk^ak. a(n) = a1*a(p1)+a2*a(p2)+...ak*a(pk). Now we focus on p^a, which is some prime power in prime factorisation of n. There are tau(n)/(a+1) divisors which don't have p as one of their divisors. It follows that there are tau(n)/(a+1) divisors with 1 as their highest power of p, tau(n)/(a+1) divisors with 2 as their highest power of p, ect up to a. So a(p1^1)*tau(n)/(a+1) +  a(p1^2)*tau(n)/(a+1) + ... = a(p) * tau(n)/(a+1) * a*(a+1)/2 = a(p)*tau(n) * a/2. Summing all p's = tau(n)/2 * Sum_{i=1,k} (ai*a(pi)) = tau(n)/2 * a(n).

So we get 2*a(n) = tau(n)/2 * a(n), tau(n) = 4. In this last step here, we divide by a(n), so thats the reason it can't be 0.

Is the proof correct? Is this already known? Should it be added as a comment to A007422?

Since I am already writing this, how to get the premission to create/edit wiki pages? I think it would be nice to have a page listing all the sequences of generalized perfect numbers (along with definiton of them), or adding them to the index page.

Best,
Žiga Pirc