Highly Composite Numbers
The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A002201 in the OEIS) are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither.
Superior highly composite number Wikiwand
A superior highly composite number is a positive integer n n for which there is an ϵ > 0 ϵ > 0 such that d(n) nϵ ≥ d(k) kϵ d ( n) n ϵ ≥ d ( k) k ϵ for all k > 1 k > 1, where the function d(n) d ( n) counts the divisors of n n.
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For a superior highly composite number n there exists a positive real number ε such that for all natural numbers k smaller than n we have. and for all natural numbers k larger than n we have. where d(n), the divisor function, denotes the number of divisors of n.The term was coined by Ramanujan (1915).. The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440.
Superior highly composite number Detailed Pedia
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).
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A highly composite number is a positive integer with more divisors than any smaller positive integer has. A related concept is that of a largely composite number, a positive integer which has at least as many divisors as any smaller positive integer.
Highly composite numbers
A superior highly composite number is a natural number which has more divisors than any other number scaled relative to the number itself. number-theory elementary-number-theory prime-numbers divisibility divisor-sum Share Cite Follow asked Sep 6, 2014 at 23:22 Omega Force 822 5 14 1
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For s=1, these numbers have been called superabundant by L. Alaoglu and P. Erdős , , and the generalized superior highly composite numbers have been called colossally abundant. The real solution of 2 s +4 s +8 s =3 s +9 s is approximately 1.6741. 10.60. For s=1, the results of this and the following section are in and . 10.62.
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In mathematics, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it's defined by a ratio between the number of divisors an integer has and that integer raised to some positive power.
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The number twelve, a superior highly composite number, is the smallest number with four non-trivial factors (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing range, and the smallest abundant number.
Superior highly composite number Alchetron, the free social encyclopedia
There are an infinite number of highly composite numbers, and the first few are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040,. (OEIS A002182 ). The corresponding numbers of divisors are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32,. (OEIS A002183 ).
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When s= 1 these numbers have been called colossally abundant by Erd}os and Alaoglu. It is easily seen that all generalized superior highly composite numbers are generalized highly composite. The prime factorization of a generalized superior highly composite number Ncan be obtained from the value of the parameter ". For r= 1;2;3:::, Ramanujan de.
Prime and composite numbers Education is the most important
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A number is superior highly composite whenever it has the largest SHCN score for some positive e - when it can be given some disadvantage with the smaller numbers and still have a higher divisor score. The first highly composite number that fails this is 24. It has 8 divisors, and for any e, 8 / 24 e makes this score smaller than for some.
Composite Numbers
In mathematics, a superior highly composite number is a certain kind of natural number. Formally, a natural number n is called superior highly composite iff there is an ε > 0 such that for all natural numbers k ≥ 1, where d(n), the divisor function, denotes the number of divisors of n. The first few superior highly composite numbers are 2, 6.
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superior highly composite number. Natural Language; Math Input; Extended Keyboard Examples Upload Random. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music…
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The first several superior highly composite numbers are 2, 6, 12, 60, 120, 360. The sequence is http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=002201A002201 in Sloane's encyclopedia. References 1L. Alaoglu and P. Erdös, On highly composite and similar numbers. Trans. Amer. Math. Soc. 56(1944), 448-469.