Are you sure about the first rule? It doesn't seem necessary according to https://oeis.org/A014575.
The problem is really fun.
Since I created the problem series and want the Functions I badge, I thought I'd provide the solution that incorporates vampire_factor and are_fangs. Not a great score, but readable!
Sorry, I mean the Magic Numbers badge.
Return the 3n+1 sequence for n
Sum of diagonal of a square matrix
Replicate elements in vectors
Sum of the Matrix Elements
Fangs of a vampire number
Find all vampire fangs
Multivariate polynomials - sort monomials
Multivariate polynomials - overload multiplication
Negation the hard way
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