Question

Interesting observation I wanted to share about perfect numbers.

Answer 1

\(\phi(n)\) is the Euler totient function, \(\tau(n)\) is the divisor counting function, and the notation at the bottom of the sum \(c|n\) means sum over all the composite divisors, c, of the number n. \[\phi (n) = \sum_{c|n}\phi(\frac{n}{c})[ \tau(c)-2]\]

This is true for all perfect numbers, whether they're even or odd.

Answer 2

Quick example:

\[\phi(6) = \phi(\frac{6}{6}) [\tau(6) -2] = 1*(4-2)=2\]

Ok so the only composite divisor of 6 is 6 itself, so not a very exciting sum, so how about a different perfect number:

\[\phi(28) = \phi(\frac{28}{4}) [\tau(4)-2] + \phi(\frac{28}{14}) [\tau(14)-2] + \phi(\frac{28}{28}) [\tau(28)-2] = 12\]

Answer 3

Ok examples done (shoddily I'll admit and poorly explained but no one's asking and just sorta pushing this out for other people who might be interested before I type this up in latex for myself, but feel free to ask anything about any aspect about this)

Proof:

Perfect numbers obey this relationship:
\[\sigma(n)=2n\]
\[\sigma(n)-2n=0\]
Two Dirichlet convolution identities:
\[\sigma = \phi \star \tau\]\[n = \phi \star u\]
Combining these with the perfect number relationship:
\[0 = \sigma-2n = \phi \star \tau - 2 \phi \star u = \phi \star (\tau -2 u)\]
Expanding it out the summation implied by the Dirichlet convolution:
\[0 = \sum_{d|n} \phi(\frac{n}{d})[\tau(d)-2] \]
Since \(\tau(1)=1\) and for any prime p = d, \(\tau(p)=2\) then we see al the prime divisors will leave this expression and the only negative one will be when \(\tau(1)-2=-1\), so I pull this term out and then relabel the divisors to be c instead of d to indicate they are only the composite divisors.

\[0 =\phi(\frac{n}{1})[\tau(1)-2]+ \sum_{c|n} \phi(\frac{n}{c})[\tau(c)-2] \]

\[\phi(n)= \sum_{c|n} \phi(\frac{n}{c})[\tau(c)-2] \]

So there it is. Maybe there's more that can be done with this, just playing around this afternoon.

Answer 4

what about the relationship shared between the two numbers 27 and 37? Eh?

Answer 5

Nice