Editing Highly composite number (section)

{{Short description|Numbers with many divisors}}
{{about|numbers having many divisors|numbers factorized only to powers of 2, 3, 5 and 7 (also named 7-smooth numbers)|Smooth number}}

[[File:Highly composite number Cuisenaire rods 6.png|75px|thumb|Demonstration, with [[Cuisenaire rods]], of the first four highly composite numbers: 1, 2, 4, 6]]

A '''highly composite number''' is a [[positive integer]] that has more [[divisors]] than all smaller positive integers. If ''d''(''n'') denotes the number of divisors of a positive integer ''n'', then a positive integer ''N'' is highly composite if ''d''(''N'') > ''d''(''n'') for all ''n'' < ''N''. For example, 6 is highly composite because ''d''(6)=4, and for ''n''=1,2,3,4,5, you get ''d''(''n'')=1,2,2,3,2, respectively, which are all less than 4. 

A related concept is that of a '''largely composite number''', a positive integer that has at least as many divisors as all smaller positive integers. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually [[composite numbers]]; however, all further terms are.

[[Ramanujan]] wrote a paper on highly composite numbers in 1915.<ref>{{cite journal | last1=Ramanujan | first1=S. | author1-link=Srinivasa Ramanujan | title=Highly composite numbers | jfm=45.1248.01 | doi=10.1112/plms/s2_14.1.347 | journal=Proc. London Math. Soc. |series=Series 2 | volume=14 | pages=347–409 | year=1915| url=http://ramanujan.sirinudi.org/Volumes/published/ram15.pdf}}<!-- https://zenodo.org/record/1433496/files/article.pdf --></ref>

The mathematician [[Jean-Pierre Kahane]] suggested that [[Plato]] must have known about highly composite numbers as he deliberately chose such a number, [[5040 (number)|5040]] (=&nbsp;[[Factorial|7!]]), as the ideal number of citizens in a city.<ref>{{citation|first=Jean-Pierre|last=Kahane|author-link=Jean-Pierre Kahane|title=Bernoulli convolutions and self-similar measures after Erdős: A personal hors d'oeuvre|journal=Notices of the American Mathematical Society|date=February 2015|volume=62|issue=2|pages=136–140}}. Kahane cites Plato's [[Laws (dialogue)|''Laws'']], 771c.</ref> Furthermore, Vardoulakis and Pugh's paper delves into a similar inquiry concerning the number 5040.<ref>{{citation|last1=Vardoulakis|first1=Antonis|last2=Pugh|first2=Clive|title=Plato's hidden theorem on the distribution of primes|journal=The Mathematical Intelligencer|date=September 2008|volume=30|issue=3|pages=61–63|doi=10.1007/BF02985381 |url=https://link.springer.com/article/10.1007/BF02985381}}.</ref>