Let σ_∞(n) be the sum of the infinitary divisors of a number n. An infinitary perfect number is a number n such that σ_∞(n) = 2n. The first few are 6, 60, 90, 36720, ... (OEIS A007357). Cohen found 14 such numbers, and 155 are known as of January 2004 (Pedersen).
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