Perfect numbers are numbers that have the property $$\sigma(n) = 2n$$ A generalization of perfect numbers are superperfect numbers, which have the property $$\sigma(\sigma(n)) = 2n$$ I wonder if there are any numbers that use $3$ or more iterations of the divisor function to equal 2n. According to Wolfram MathWorld, there are no even numbers satisfying this property. Is it possible to prove this for odd numbers as well, or do these generalized superperfect numbers exist?
2026-03-25 06:02:32.1774418552
Are there any $\ m\ $- superperfect numbers for $\ \ m>2\ \ $?
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