Editing Highly composite number (section)

==Related sequences==
{{Euler_diagram_numbers_with_many_divisors.svg}}
Highly composite numbers greater than 6 are also [[abundant numbers]]. One need only look at the three largest proper divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also [[Harshad numbers]] in base 10. The first highly composite number that is not a Harshad number is 245,044,800; it has a digit sum of 27, which does not divide evenly into 245,044,800.

10 of the first 38 highly composite numbers are [[superior highly composite numbers]].
The sequence of highly composite numbers {{OEIS|id=A002182}} is a subset of the sequence of smallest numbers ''k'' with exactly ''n'' divisors {{OEIS|id=A005179}}.

Highly composite numbers whose number of divisors is also a highly composite number are
: 1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200 {{OEIS|id=A189394}}.
It is extremely likely that this sequence is complete.

A positive integer ''n'' is a '''largely composite number''' if ''d''(''n'') ≥ ''d''(''m'') for all ''m'' ≤ ''n''.  The counting function ''Q''<sub>''L''</sub>(''x'') of largely composite numbers satisfies
:<math>(\log x)^c \le \log Q_L(x) \le (\log x)^d \ </math>
for positive ''c'' and ''d'' with <math>0.2 \le c \le d \le 0.5</math>.<ref name=HNTI46>Sándor et al. (2006) p.&nbsp;46</ref><ref name=Nic79>{{cite journal | last=Nicolas | first=Jean-Louis | author-link=Jean-Louis Nicolas | title=Répartition des nombres largement composés | language=fr | zbl=0368.10032 | journal=Acta Arith. | volume=34 | issue=4 | pages=379–390 | year=1979 | doi=10.4064/aa-34-4-379-390 | doi-access=free}}</ref>

Because the prime factorization of a highly composite number uses all of the first ''k'' primes, every highly composite number must be a [[practical number]].<ref>{{citation
 | last = Srinivasan | first = A. K.
 | title = Practical numbers
 | journal = [[Current Science]]
 | volume = 17
 | year = 1948
 | pages = 179–180
 | mr=0027799
 | url = http://www.ias.ac.in/jarch/currsci/17/179.pdf}}.</ref> Due to their ease of use in calculations involving [[Fraction (mathematics)|fractions]], many of these numbers are used in [[Historical weights and measures|traditional systems of measurement]] and engineering designs.