For some reason, fixed points in multiplicative functions fascinate me. I was thinking about one of my favorite functions, the divisor function (sigma), and figured out an interesting minor mathematical tidbit, which I’m recording now so I don’t forget it.

If x is an (even) perfect number, x = 2^m-1 * (2^m – 1), where m is a Mersenne prime.
σ(x) = 2x by definition, but σ(σ(x)), that is, σ(2x) = 2^m * (2^m+1 – 1).