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On Disreputable Numbers

One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery — and more totally useless — than the perfect numbers.

— Martin Gardner

There are countless ways to classify integers. Happy, perfect, friendly, sociable, abundant, extravagant, cute, interesting, frugal, deficient, hungry, undulating, weird, vampire… the list goes on. But how useful are such classifications, beyond their inherent interestingness, and as a hook to get people into number theory?