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where σ denotes the divisor function (i.e., the sum of all positive divisors of n, including n itself). The first few superabundant numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ... ; superabundant numbers are closely related to highly composite numbers.
Superabundant numbers were first defined in [AlaErd44].
Alaoglu and Erdős proved [AlaErd44] that if n is superabundant, then there exist a2, ..., ap such that
In fact, ap is nearly always 1.
It can also be shown that all superabundant numbers are Harshad numbers.
- [AlaErd44] - Leonidas Alaoglu and Paul Erdős, On Highly Composite and Similar Numbers, Trans. AMS 56, 448-469 (1944)
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