Superior highly composite number

Divisor function d(n) up to n = 250

Prime-power factors

In mathematics, a superior highly composite number is a natural number which has more divisors than any other number scaled relative to some power of the number itself. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer.

The first 10 superior highly composite numbers and their factorization are listed.

# prime factors	SHCN n	prime factorization	prime exponents	# divisors d(n)		primorial factorization
1	2	$2$	1	2	2	$2$
2	6	$2 \cdot 3$	1,1	2²	4	$6$
3	12	$22 \cdot 3$	2,1	3×2	6	$2 \cdot 6$
4	60	$22 \cdot 3 \cdot 5$	2,1,1	3×2²	12	$2 \cdot 30$
5	120	$23 \cdot 3 \cdot 5$	3,1,1	4×2²	16	$22 \cdot 30$
6	360	$23 \cdot 3 2 \cdot 5$	3,2,1	4×3×2	24	$2 \cdot 6 \cdot 30$
7	2520	$23 \cdot 3 2 \cdot 5 \cdot 7$	3,2,1,1	4×3×2²	48	$2 \cdot 6 \cdot 210$
8	5040	$24 \cdot 3 2 \cdot 5 \cdot 7$	4,2,1,1	5×3×2²	60	$22 \cdot 6 \cdot 210$
9	55440	$24 \cdot 3 2 \cdot 5 \cdot 7 \cdot 11$	4,2,1,1,1	5×3×2³	120	$22 \cdot 6 \cdot 2310$
10	720720	$24 \cdot 3 2 \cdot 5 \cdot 7 \cdot 11 \cdot 13$	4,2,1,1,1,1	5×3×2⁴	240	$22 \cdot 6 \cdot 30030$

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

\frac{d(n)}{n^\varepsilon}\geq\frac{d(k)}{k^\varepsilon}

and for all natural numbers k larger than n we have

{\frac {d(n)}{n^{\varepsilon }}}>{\frac {d(k)}{k^{\varepsilon }}}

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, 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.

Properties

All superior highly composite numbers are highly composite.

An effective construction of the set of all superior highly composite numbers is given by the following monotonic mapping from the positive real numbers.^[1] Let

e_p(x) = \left\lfloor \frac{1}{\sqrt[x]{p} - 1} \right\rfloor\quad

for any prime number p and positive real x. Then

\quad s(x) = \prod_{p \in \mathbb{P}} p^{e_p(x)}\quad

is a superior highly composite number.

Note that the product need not be computed indefinitely, because if $p > 2^x$ then $e_p(x) = 0$ , so the product to calculate $s(x)$ can be terminated once $p \ge 2^x$ .

Also note that in the definition of $e_p(x)$ , $1/x$ is analogous to $\varepsilon$ in the implicit definition of a superior highly composite number.

Moreover for each superior highly composite number $s^{\prime }$ exists a half-open interval $I \subset \R^+$ such that $\forall x \in I: s(x) = s^\prime$ .

This representation implies that there exist an infinite sequence of $\pi_1, \pi_2, \ldots \in \mathbb{P}$ such that for the n-th superior highly composite number $s_{n}$ holds

s_n = \prod_{i=1}^n\pi_i

The first $\pi _{i}$ are 2, 3, 2, 5, 2, 3, 7, ... (sequence A000705 in the OEIS). In other words, the quotient of two successive superior highly composite numbers is a prime number.

Divisors

The divisors of the first ten superior highly composite numbers are:

2: 1, 2
6: 1, 2, 3, 6
12: 1, 2, 3, 4, 6, 12
60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
2520: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520
5040: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040
55440: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 48, 55, 56, 60, 63, 66, 70, 72, 77, 80, 84, 88, 90, 99, 105, 110, 112, 120, 126, 132, 140, 144, 154, 165, 168, 176, 180, 198, 210, 220, 231, 240, 252, 264, 280, 308, 315, 330, 336, 360, 385, 396, 420, 440, 462, 495, 504, 528, 560, 616, 630, 660, 693, 720, 770, 792, 840, 880, 924, 990, 1008, 1155, 1232, 1260, 1320, 1386, 1540, 1584, 1680, 1848, 1980, 2310, 2520, 2640, 2772, 3080, 3465, 3696, 3960, 4620, 5040, 5544, 6160, 6930, 7920, 9240, 11088, 13860, 18480, 27720, 55440
720720: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 52, 55, 56, 60, 63, 65, 66, 70, 72, 77, 78, 80, 84, 88, 90, 91, 99, 104, 105, 110, 112, 117, 120, 126, 130, 132, 140, 143, 144, 154, 156, 165, 168, 176, 180, 182, 195, 198, 208, 210, 220, 231, 234, 240, 252, 260, 264, 273, 280, 286, 308, 312, 315, 330, 336, 360, 364, 385, 390, 396, 420, 429, 440, 455, 462, 468, 495, 504, 520, 528, 546, 560, 572, 585, 616, 624, 630, 660, 693, 715, 720, 728, 770, 780, 792, 819, 840, 858, 880, 910, 924, 936, 990, 1001, 1008, 1040, 1092, 1144, 1155, 1170, 1232, 1260, 1287, 1320, 1365, 1386, 1430, 1456, 1540, 1560, 1584, 1638, 1680, 1716, 1820, 1848, 1872, 1980, 2002, 2145, 2184, 2288, 2310, 2340, 2520, 2574, 2640, 2730, 2772, 2860, 3003, 3080, 3120, 3276, 3432, 3465, 3640, 3696, 3960, 4004, 4095, 4290, 4368, 4620, 4680, 5005, 5040, 5148, 5460, 5544, 5720, 6006, 6160, 6435, 6552, 6864, 6930, 7280, 7920, 8008, 8190, 8580, 9009, 9240, 9360, 10010, 10296, 10920, 11088, 11440, 12012, 12870, 13104, 13860, 15015, 16016, 16380, 17160, 18018, 18480, 20020, 20592, 21840, 24024, 25740, 27720, 30030, 32760, 34320, 36036, 40040, 45045, 48048, 51480, 55440, 60060, 65520, 72072, 80080, 90090, 102960, 120120, 144144, 180180, 240240, 360360, 720720

Superior highly composite radices

The first few superior highly composite numbers have often been used as radices, due to their high divisibility for their size. For example:

Binary (base 2)
Senary (base 6)
Duodecimal (base 12)
Sexagesimal (base 60)

120 appears as the long hundred, while 360 appears as the number of degrees in a circle.

Notes

↑ Ramanujan (1915); see also URL http://wwwhomes.uni-bielefeld.de/achim/hcn.dvi

References

Ramanujan, S. (1915). "Highly composite numbers". Proc. London Math. Soc. (2). 14: 347–409. doi:10.1112/plms/s2_14.1.347. JFM 45.1248.01. Reprinted in Collected Papers (Ed. G. H. Hardy et al.), New York: Chelsea, pp. 78–129, 1962
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.

External links

Weisstein, Eric W. "Superior highly composite number". MathWorld.

Divisibility-based sets of integers

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