Matches in DBpedia 2014 for { <http://dbpedia.org/resource/Highly_composite_number> ?p ?o. }
Showing items 1 to 48 of
48
with 100 items per page.
- Highly_composite_number abstract "A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. The term was coined by Ramanujan (1915), who showed that there are infinitely many such numbers. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The initial or smallest twenty-six highly composite numbers are listed in the table at right. The sequence of highly composite numbers (sequence A002182 in OEIS) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 in OEIS).Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, but not too many of the same. If we decompose a number n in prime factors like this:where are prime, and the exponents are positive integers, then the number of divisors of n is exactlyHence, for n to be a highly composite number, the k given prime numbers pi must be precisely the first k prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than n with the same number of divisors (for instance 10 = 2 × 5 may be replaced with 6 = 2 × 3; both have four divisors); the sequence of exponents must be non-increasing, that is otherwise, by exchanging two exponents we would again get a smaller number than n with the same number of divisors (for instance 18 = 21 × 32 may be replaced with 12 = 22 × 31; both have six divisors).Also, except in two special cases n = 4 and n = 36, the last exponent ck must equal 1. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials. 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.Highly composite numbers higher than 6 are also abundant numbers. One need only look at the three or four highest 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 HCN that is not a Harshad number is 245,044,800, which has a digit sum of 27, but 27 does not divide evenly into 245,044,800.Many of these numbers are used in traditional systems of measurement, and tend to be used in engineering designs, due to their ease of use in calculations involving fractions.If Q(x) denotes the number of highly composite numbers less than or equal to x, then there are two constants a and b, both greater than 1, such thatThe first part of the inequality was proved by Paul Erdős in 1944 and the second part by Jean-Louis Nicolas in 1988. We haveand".
- Highly_composite_number thumbnail Highly_composite_numbers.svg?width=300.
- Highly_composite_number wikiPageExternalLink hcn-algorithm.tex.
- Highly_composite_number wikiPageExternalLink hcn10000.txt.gz.
- Highly_composite_number wikiPageExternalLink highlycompositenumbers.htm.
- Highly_composite_number wikiPageExternalLink highly.html.
- Highly_composite_number wikiPageID "208732".
- Highly_composite_number wikiPageRevisionID "604581543".
- Highly_composite_number hasPhotoCollection Highly_composite_number.
- Highly_composite_number title "Highly Composite Number".
- Highly_composite_number urlname "HighlyCompositeNumber".
- Highly_composite_number subject Category:Conjectures.
- Highly_composite_number subject Category:Integer_sequences.
- Highly_composite_number type Abstraction100002137.
- Highly_composite_number type Arrangement107938773.
- Highly_composite_number type Cognition100023271.
- Highly_composite_number type Concept105835747.
- Highly_composite_number type Conjectures.
- Highly_composite_number type Content105809192.
- Highly_composite_number type Group100031264.
- Highly_composite_number type Hypothesis105888929.
- Highly_composite_number type Idea105833840.
- Highly_composite_number type IntegerSequences.
- Highly_composite_number type Ordering108456993.
- Highly_composite_number type PsychologicalFeature100023100.
- Highly_composite_number type Sequence108459252.
- Highly_composite_number type Series108457976.
- Highly_composite_number type Speculation105891783.
- Highly_composite_number comment "A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. The term was coined by Ramanujan (1915), who showed that there are infinitely many such numbers. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The initial or smallest twenty-six highly composite numbers are listed in the table at right.".
- Highly_composite_number label "Highly composite number".
- Highly_composite_number label "Hochzusammengesetzte Zahl".
- Highly_composite_number label "Hogelijk samengesteld getal".
- Highly_composite_number label "Nombre hautement composé".
- Highly_composite_number label "Numero altamente composto".
- Highly_composite_number label "高合成数".
- Highly_composite_number label "高度合成数".
- Highly_composite_number sameAs Hochzusammengesetzte_Zahl.
- Highly_composite_number sameAs Nombre_hautement_composé.
- Highly_composite_number sameAs Numero_altamente_composto.
- Highly_composite_number sameAs 高度合成数.
- Highly_composite_number sameAs Hogelijk_samengesteld_getal.
- Highly_composite_number sameAs m.01dn7t.
- Highly_composite_number sameAs Q629958.
- Highly_composite_number sameAs Q629958.
- Highly_composite_number sameAs Highly_composite_number.
- Highly_composite_number wasDerivedFrom Highly_composite_number?oldid=604581543.
- Highly_composite_number depiction Highly_composite_numbers.svg.
- Highly_composite_number isPrimaryTopicOf Highly_composite_number.