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Superior Highly Composite Number


A superior highly composite number is a positive integer n for which there is an e>0 such that

 (d(n))/(n^e)>=(d(k))/(k^e)

for all k>1, where the function d(n) counts the divisors of n (Ramanujan 1962, pp. 87 and 115). It can be shown that all superior highly composite numbers are highly composite and that the nth superior highly composite number has the form pi_1pi_2pi_3...pi_n, where the factors pi_k are prime.

The first few superior highly composite numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, ... (OEIS A002201), and the corresponding sequence of primes pi_k is 2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, 7, 29, 3, 31, 2, 37, 41, 43, ... (OEIS A000705).


See also

Abundant Number, Colossally Abundant Number, Highly Composite Number, Round Number, Smooth Number

This entry contributed by David Terr

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References

Ramanujan, S. "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.Sloane, N. J. A. Sequences A000705/M0423 and A002201/M1591 in "The On-Line Encyclopedia of Integer Sequences."

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Superior Highly Composite Number

Cite this as:

Terr, David. "Superior Highly Composite Number." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SuperiorHighlyCompositeNumber.html

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