Superior highly composite number
where d(n), the divisor function, denotes the number of divisors of n. The first superior highly composite numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200... (sequence A002201 in OEIS).
All superior highly composite numbers are highly composite.
An effective construction of the set of all superior highly composite numbers is given by the following monotonic mapping from the positive real numbers. Let
is a superior highly composite number.
Notice: If then . Therefore for each fixed only finite many are non-zero and the just defined product for has only finite many factors unequal 1. This in the definition of is the in the implicit definition of a superior highly composite number.
Moreover for each superior highly composite number exists a half-open interval such that .
This representation implies that there exist an infinite sequence of such that for the n-th superior highly composite number holds
- Srinivasa Ramanujan, Highly Composite Numbers, Proc. London Math. Soc. 14, 347-407, 1915; reprinted in Collected Papers (Ed. G. H. Hardy et al.), New York: Chelsea, pp. 78–129, 1962
- Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 45–46. ISBN 1-4020-4215-9. Zbl 1151.11300.
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