For every $k>2$, does there exist at least one k-perfect number?

Question

For every $k>2$, does there exist at least one number $n$ for which $\sigma(n)=k*n$? What kind of heuristics can be done to shed light on this problem?

Answer 1

See Wikipedia's k-perfect numbers:

In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.

For a given natural number k, a number n is called k-perfect (or k-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number

$$ \small \begin{array}{rrl} \\ k & \text{Smallest k-perfect number} & \\ \hline \\ 2 & 6 & 2 × 3 \\ 3 & 120 & 2^3 × 3 × 5 \\ 4 & 30240 & 2^5 × 3^3 × 5 × 7 \\ 5 & 14182439040 & 2^7 × 3^4 × 5 × 7 × 11^2 × 17 × 19 \\ 6 & \text{(21 digits)} \, 154345556085770649600 & 2^15 × 3^5 × 5^2 × 7^2 × 11 × 13 × 17 × 19 × 31 × 43 × 257 \\ 7 & ... & \\ \end{array} $$

See the Wikipedia page linked above for k-perfect numbers up to 11, they are too lengthy to fit on Stack Exchange pages. It's reasonable to expect that there are more but they haven't been discovered yet.

Other references:

Oeis - Multiply-Perfect Numbers

MathWorld - Multiperfect Number

"Moxham (2000) found the largest known multiperfect number, approximately equal to 7.3×10^(1345), on Feb. 13, 2000."

The "The Multiply Perfect Numbers Page" has a gzipped file from 2014 with 5311 MPNs, as of 2018-01-07 no new numbers have been discovered.