Highly composite number

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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 38 highly composite numbers are listed in the table at right. 10 of them are superior highly composite numbers.

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. By the fundamental theorem of arithmetic, every positive integer n has a unique prime factorization:

${\displaystyle n=p_{1}^{c_{1}}\times p_{2}^{c_{2}}\times \cdots \times p_{k}^{c_{k}}\qquad (1)}$

where ${\displaystyle p_{1} are prime, and the exponents ${\displaystyle c_{i}}$ are positive integers.

Any factor of n must have the same or lesser multiplicity in each prime:

${\displaystyle p_{1}^{d_{1}}\times p_{2}^{d_{2}}\times \cdots \times p_{k}^{d_{k}},0\leq d_{i}\leq c_{i},0

So the number of divisors of n is:

${\displaystyle d(n)=(c_{1}+1)\times (c_{2}+1)\times \cdots \times (c_{k}+1).\qquad (2)}$

Hence, 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);

Also, except in two special cases n = 4 and n = 36, the last exponent ck must equal 1. It means that 1, 4, and 36 are the only square highly composite numbers. 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.[1]

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 that

${\displaystyle \ln(x)^{a}\leq Q(x)\leq \ln(x)^{b}\,.}$

The first part of the inequality was proved by Paul Erdős in 1944 and the second part by Jean-Louis Nicolas in 1988. We have[2]

${\displaystyle 1.13862<\liminf {\frac {\log Q(x)}{\log \log x}}\leq 1.44\ }$

and

${\displaystyle \limsup {\frac {\log Q(x)}{\log \log x}}\leq 1.71\ .}$

Examples

 The highly composite number: 10080 10080 = (2 × 2 × 2 × 2 × 2)  ×  (3 × 3) ×  5  ×  7 By (2) above, 10080 has exactly seventy-two divisors. 1×10080 2 × 5040 3 × 3360 4 × 2520 5 × 2016 6 × 1680 7× 1440 8 × 1260 9 × 1120 10 × 1008 12 × 840 14 × 720 15× 672 16 × 630 18 × 560 20 × 504 21 × 480 24 × 420 28× 360 30 × 336 32 × 315 35 × 288 36 × 280 40 × 252 42× 240 45 × 224 48 × 210 56 × 180 60 × 168 63 × 160 70× 144 72 × 140 80 × 126 84 × 120 90 × 112 96 × 105 Note:  Numbers in bold are themselves highly composite numbers. Only the twentieth highly composite number 7560 (= 3 × 2520) is absent.10080 is a so-called 7-smooth number (sequence A002473 in OEIS).

The 15,000th highly composite number can be found on Achim Flammenkamp's website. It is the product of 230 primes:

${\displaystyle a_{0}^{14}a_{1}^{9}a_{2}^{6}a_{3}^{4}a_{4}^{4}a_{5}^{3}a_{6}^{3}a_{7}^{3}a_{8}^{2}a_{9}^{2}a_{10}^{2}a_{11}^{2}a_{12}^{2}a_{13}^{2}a_{14}^{2}a_{15}^{2}a_{16}^{2}a_{17}^{2}a_{18}^{2}a_{19}a_{20}a_{21}\cdots a_{229},}$

where ${\displaystyle a_{n}}$ is the sequence of successive prime numbers, and all omitted terms (a22 to a228) are factors with exponent equal to one (i.e. the number is ${\displaystyle 2^{14}\times 3^{9}\times 5^{6}\times \cdots \times 1451}$). [3]

Plot of the number of divisors of integers from 1 to 1000. The first 15 highly composite numbers are in bold.

Prime factor subsets

For any highly composite number, if one takes any subset of prime factors for that number and their exponents, the resulting number will have more divisors than any smaller number that uses the same prime factors. For example for the highly composite number 720 which is 24 × 32 × 5 we can be sure that

• 144 which is 24 × 32 has more divisors than any smaller number that has only the prime factors 2 and 3
• 80 which is 24 × 5 has more divisors than any smaller number that has only the prime factors 2 and 5
• 45 which is 32 × 5 has more divisors than any smaller number that has only the prime factors 3 and 5

If this were untrue for any particular highly composite number and subset of prime factors, we could exchange that subset of prime factors and exponents for the smaller number using the same prime factors and get a smaller number with at least as many divisors.

This property is useful for finding highly composite numbers.

Largely composite numbers

A positive integer n is a largely composite number if d(n) ≥ d(m) for all mn. The counting function QL(x) of largely composite numbers satisfies

${\displaystyle (\log x)^{c}\leq \log Q_{L}(x)\leq (\log x)^{d}\ }$

Notes

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2. Sándor et al (2006) p.45
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4. Sándor et al (2006) p.46
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References

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