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Highly abundant number
In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number.
Highly abundant numbers and several similar classes of numbers were first introduced by Pillai (1943), and early work on the subject was done by Alaoglu and Erdős (1944). Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any N is at least proportional to log2 N. They also proved that 7200 is the largest powerful highly abundant number, and therefore the largest highly abundant number with odd sum of divisors.
Formal definition and examples
Formally, a natural number n is called highly abundant if and only if for all natural numbers m < n,
\( \sigma(n) > \sigma(m) \)
where σ denotes the sum-of-divisors function. The first few highly abundant numbers are
1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... (sequence A002093 in OEIS).
For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ.
Relations with other sets of numbers
Some sources[citation needed] report that all factorials are highly abundant numbers, but this is incorrect.
σ(9!) = σ(362880) = 1481040,
but there is a smaller number with larger sum of divisors,
σ(360360) = 1572480,
so 9! is not highly abundant.
Alaoglu and Erdős noted that all superabundant numbers are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant. This question was answered affirmatively by Nicolas (1969).
Despite the terminology, not all highly abundant numbers are abundant numbers. In particular, none of the first seven highly abundant numbers is abundant.
References
Alaoglu, L.; Erdős, P. (1944). "On highly composite and similar numbers". Transactions of the American Mathematical Society 56 (3): 448–469. doi:10.2307/1990319. JSTOR 1990319. MR0011087.
Nicolas, Jean-Louis (1969). "Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers"". Bull. Soc. Math. France 97: 129–191. MR0254130.
Pillai, S. S. (1943). "Highly abundant numbers". Bull. Calcutta Math. Soc. 35: 141–156. MR0010560.
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