Superabundant Number

A superabundant number is a composite number  such that  for all , where  is the divisor function. Superabundant numbers are closely related to highly composite numbers, and the first 19 superabundant and highly composite numbers are the same. The first few are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, ... (OEIS A004394).

REFERENCES:

Alaoglu, L. and Erdős, P. "On Highly Composite and Similar Numbers." Trans. Amer. Math. Soc. 56, 448-469, 1944.

Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., p. 112, 1973.

Honsberger, R. "An Introduction to Ramanujan's Highly Composite Numbers." Ch. 14 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 193-207, 1985.

Sloane, N. J. A. Sequence A004394 in "The On-Line Encyclopedia of Integer Sequences."

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. New York: Penguin Books, pp. 85 and 128, 1986.