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.
Find relatively common elements in matrix rows
Whether the input is vector?
Vectorize the digits of an Integer
5 Prime Numbers
Fangs of a vampire number
Multivariate polynomials - emulate symbolic form
Multivariate polynomials - sort monomials
Negation the hard way
Multivariate polynomials - convert monomial form to array
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