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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.
Highly composite numbers
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A superior highly composite number is a natural number which has more divisors than any other
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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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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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.
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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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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.
Composite Numbers
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.
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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
Highly composite numbers
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.
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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.
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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.
Superior highly composite number Wikiwand
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 ).
Highly Composite Numbers
In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power.
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1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160 ( list ; graph ; refs ; listen ; history ; text ; internal format )