kb/data/en.wikipedia.org/wiki/Highly_composite_number-1.md

title	chunk	source	category	tags	date_saved	instance
Highly composite number	2/2	https://en.wikipedia.org/wiki/Highly_composite_number	reference	science, encyclopedia	2026-05-05T08:15:00.548879+00:00	kb-cron

      lim sup
      
        x
        
        →
        
        ∞
      
    
    
      
        
          log
          ⁡
          Q
          (
          x
          )
        
        
          log
          ⁡
          log
          ⁡
          x
        
      
    
    ≤
    1.71
     
    .
  

{\displaystyle \limsup _{x\,\to \,\infty }{\frac {\log Q(x)}{\log \log x}}\leq 1.71\ .}

== Related sequences ==

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 (sequence A002182 in the OEIS) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 in the OEIS). 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 (sequence A189394 in the OEIS). It is known 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 QL(x) of largely composite numbers satisfies

    (
    log
    ⁡
    x
    
      )
      
        c
      
    
    ≤
    log
    ⁡
    
      Q
      
        L
      
    
    (
    x
    )
    ≤
    (
    log
    ⁡
    x
    
      )
      
        d
      
    
     
  

{\displaystyle (\log x)^{c}\leq \log Q_{L}(x)\leq (\log x)^{d}\ }

for positive c and d with

    0.2
    ≤
    c
    ≤
    d
    ≤
    0.5
  

{\displaystyle 0.2\leq c\leq d\leq 0.5}

. 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. Due to their ease of use in calculations involving fractions, many of these numbers are used in traditional systems of measurement and engineering designs.

== See also == Superior highly composite number Highly totient number Table of divisors Euler's totient function Round number Smooth number

== Notes ==

== References == Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 45–46. ISBN 1-4020-4215-9. Zbl 1151.11300. Erdös, P. (1944). "On highly composite numbers" (PDF). Journal of the London Mathematical Society. Second Series. 19 (75_Part_3): 130–133. doi:10.1112/jlms/19.75_part_3.130. MR 0013381. Alaoglu, L.; Erdös, P. (1944). "On highly composite and similar numbers" (PDF). Transactions of the American Mathematical Society. 56 (3): 448–469. doi:10.2307/1990319. JSTOR 1990319. MR 0011087. Ramanujan, Srinivasa (1997). "Highly composite numbers" (PDF). Ramanujan Journal. 1 (2): 119–153. doi:10.1023/A:1009764017495. MR 1606180. S2CID 115619659. Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin.

== External links == Weisstein, Eric W. "Highly Composite Number". MathWorld. Algorithm for computing Highly Composite Numbers First 10000 Highly Composite Numbers as factors Achim Flammenkamp, First 779674 HCN with sigma, tau, factors Online Highly Composite Numbers Calculator 5040 and other Anti-Prime Numbers - Dr. James Grime by Dr. James Grime for Numberphile