Highly composite numbers are numbers such that divisor function Superabundant numbers
are closely related to highly composite numbers, and the first 19 superabundant and
highly composite numbers are the same.
There are an infinite number of highly composite numbers, and the first few are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040,
... (OEIS A002182 ). The corresponding numbers
of divisors are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, ... (OEIS A002183 ). Ramanujan (1915) listed 102 highly composite
numbers up to 6746328388800, but omitted 293318625600. Robin (1983) gives the first
5000 highly composite numbers, and a comprehensive survey is given by Nicholas (1988).
Flammenkamp gives a list of the first 779674 highly composite numbers.
If
(1)
is the prime factorization of a highly composite
number, then
1. The primes 2, 3, ..., primes ,
2. The exponents are nonincreasing, so
3. The final exponent
Let
(2)
Alaoglu and Erdős (1944) showed that there exists a constant
(3)
Nicholas proved that there exists a constant
(4)
See also Abundant Number ,
Divisor ,
Divisor Function ,
Highly
Cototient Number ,
Round Number ,
Roundness ,
Smooth Number ,
Superabundant
Number ,
Superior Highly Composite
Number
Explore with Wolfram|Alpha
References Alaoglu, L. and Erdős, P. "On Highly Composite and Similar Numbers." Trans. Amer. Math. Soc. 56 , 448-469, 1944. Andree,
R. V. "Ramanujan's Highly Composite Numbers." Abacus 3 ,
61-62, 1986. Berndt, B. C. Ramanujan's
Notebooks, Part IV. Dickson,
L. E. History
of the Theory of Numbers, Vol. 1: Divisibility and Primality. Flammenkamp, A. "Highly Composite Numbers."
http://wwwhomes.uni-bielefeld.de/achim/highly.html . Hoffman,
P. The
Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical
Truth. Honsberger,
R. Mathematical
Gems I. Honsberger,
R. "An Introduction to Ramanujan's Highly Composite Numbers." Ch. 14
in Mathematical
Gems III. Kanigel,
R. The
Man Who Knew Infinity: A Life of the Genius Ramanujan. Nicholas, J.-L. "On Highly Composite
Numbers." In Ramanujan
Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign,
June 1-5, 1987 Ramanujan,
S. "Highly Composite Numbers." Proc. London Math. Soc. 14 ,
347-409, 1915. Ramanujan, S. Collected
Papers of Srinivasa Ramanujan Robin,
G. "Méthodes d'optimalisation pour un problème de théories
des nombres." RAIRO Inform. Théor. 17 , 239-247, 1983. Séroul,
R. "Highly Composite Numbers." §8.14 in Programming
for Mathematicians. Siano,
D. "Highly Composite Numbers: How Can We Calculate Them?" http://www.eclipse.net/~dimona/juliannum.html . Siano,
D. B. and Siano, J. D. "An Algorithm for Generating Highly Composite
Numbers." http://wwwhomes.uni-bielefeld.de/achim/julianmanuscript3.pdf .
October 7, 1994. Sloane, N. J. A. Sequences A002182 /M1025
and A002183 /M0546 in "The On-Line Encyclopedia
of Integer Sequences." Wells, D. The
Penguin Dictionary of Curious and Interesting Numbers. Referenced on Wolfram|Alpha Highly Composite Number
Cite this as:
Weisstein, Eric W. "Highly Composite Number."
From MathWorld https://mathworld.wolfram.com/HighlyCompositeNumber.html
Subject classifications
(i.e., the number of divisors of 
) is greater than for any smaller 
. Superabundant numbers
are closely related to highly composite numbers, and the first 19 superabundant and
highly composite numbers are the same.
form a string of consecutive primes,
, and
is always 1, except for the two cases 
and 
, where it is 2.
be the number of highly composite numbers 
. Ramanujan (1915) showed that
such that
such that
