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Highly Composite Number

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A highly composite number is a positive integer n whose divisor function d(n)=sigma_0(n) (the number of divisors of n) is greater than d(m) for every positive integer m<n. Superabundant numbers are closely related to highly composite numbers, and the first 19 superabundant numbers 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, pp. 358-360) listed 102 highly composite numbers up to 6746328388800 in a table beginning with 2. The table does not count 1 and omits 293318625600. Including both gives 104 highly composite numbers up to 6746328388800. Robin (1983) computed the first 5284 highly composite numbers and used the first 5000 in tests of conjectures. A comprehensive survey is given by Nicolas (1988). Flammenkamp gives a list of the first 779674 highly composite numbers.

If

N=2^(a_2)3^(a_3)...p^(a_p)
(1)

is the prime factorization of a highly composite number N>1, then

1. The primes 2, 3, ..., p form a string of consecutive primes,

2. The exponents are nonincreasing, so a_2>=a_3>=...>=a_p, and

3. The exponent a_p of the greatest prime factor p is always 1, except for the two cases N=4=2^2 and N=36=2^2·3^2, where it is 2.

Let Q(x) be the number of highly composite numbers not exceeding x. Ramanujan (1915) showed that

lim_(x->infty)(Q(x))/(lnx)=infty.
(2)

Erdős (1944) showed that there exists a constant c_1>0 such that

Q(x)>=(lnx)^(1+c_1)
(3)

for sufficiently large x. Nicolas (1971) proved that there exists a constant c_2>0 such that

Q(x)<<(lnx)^(c_2)
(4)

for sufficiently large x.

See also

Abundant Number, Divisor, Divisor Function, Highly Cototient Number, Round Number, Roundness, Smooth Number, Superabundant Number, Superior Highly Composite Number

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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. New York: Springer-Verlag, p. 53, 1994.Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, p. 323, 2005.Erdős, P. "On Highly Composite Numbers." J. London Math. Soc. 19, 130-133, 1944. https://doi.org/10.1112/jlms/19.75_Part_3.130.Flammenkamp, A. "Highly Composite Numbers." https://wwwhomes.uni-bielefeld.de/achim/HCN/.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, pp. 88-91, 1998.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.Kanigel, R. The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Washington Square Press, p. 232, 1991.Nicolas, J.-L. "On Highly Composite Numbers." In Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987 (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin). Boston, MA: Academic Press, pp. 215-244, 1988.Nicolas, J.-L. "Répartition des nombres hautement composés de Ramanujan." Canad. J. Math. 23, 116-130, 1971. https://doi.org/10.4153/CJM-1971-012-6.Numberphile. "5040 and Other Anti-Prime Numbers." Jul. 6, 2016. https://www.youtube.com/watch?v=2JM2oImb9Qg.Ramanujan, S. "Highly Composite Numbers." Proc. London Math. Soc. 14, 347-409, 1915.Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., 2000.Robin, G. "Méthodes d'optimisation pour un problème de théorie des nombres." RAIRO Inform. Théor. 17, 239-247, 1983.Séroul, R. "Highly Composite Numbers." §8.14 in Programming for Mathematicians. Berlin, Germany: Springer-Verlag, pp. 208-213, 2000.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." October 7, 1994. http://wwwhomes.uni-bielefeld.de/achim/julianmanuscript3.pdf.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. New York: Penguin Books, p. 128, 1986.

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Highly Composite Number

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Weisstein, Eric W. "Highly Composite Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HighlyCompositeNumber.html

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