A highly composite number is a positive integer N with more divisors than any positive integer smaller than it, formally defined such that the divisor function d(N') < d(N) for all N' < N, where d(k) denotes the number of positive divisors of k.[1] This concept was introduced by the Indian mathematician Srinivasa Ramanujan in a 1915 paper published in the Proceedings of the London Mathematical Society, where he systematically analyzed their structure and properties to investigate the maximal order of the divisor function.[1]Highly composite numbers exhibit a specific canonical form in their prime factorization: they are products of the smallest primes raised to non-increasing exponents, typically N = 2^{a_2} \cdot 3^{a_3} \cdot 5^{a_5} \cdots p^{a_p} where a_2 \geq a_3 \geq a_5 \geq \cdots \geq a_p \geq 1, and the exponents generally decrease strictly except possibly in trailing groups of equal values.[1] Ramanujan proved that the exponents must follow this decreasing pattern, with the final exponent usually being 1 (with rare exceptions like 4 and 36, where it is 2), ensuring that these numbers maximize the divisor count relative to their size.[1] He further established bounds on the growth of d(N), showing that it is asymptotically O\left( N^{\frac{c \log \log N}{\log N}} \right) for some constant c, and that d(N) < N^\delta for any \delta > 0 and sufficiently large N.[1]The sequence begins with 1 (which has d(1) = 1), followed by 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, and so on; Ramanujan's table commences with 2 and lists 102 such numbers up to one exceeding $10^{12} with d(N) = 10{,}080.[1] Among these, a subclass called superior highly composite numbers (also defined by Ramanujan) are those where the exponents satisfy stricter conditions for optimality, such as 2, 6, 12, 60, 120, and 840, which play a key role in determining the overall structure of highly composite numbers.[1] These numbers often coincide with highly factorable forms like factorials or primorials, reflecting their abundance in divisors.[1]Ramanujan's work laid the foundation for later studies in analytic number theory, including extensions by mathematicians like Paul Erdős on the distribution and asymptotic behavior of these numbers, highlighting their importance in understanding how divisor counts evolve with magnitude.[1]
Definition and Basics
Formal Definition
A positive integer n is said to be highly composite if for every positive integer m < n, the number of positive divisors d(n) satisfies d(n) > d(m). This strict inequality implies that highly composite numbers form a sequence where each term achieves a new maximum value for the divisor function up to that point, beginning with n = 1 where d(1) = 1.[1]The divisor function d(n) simply counts the positive divisors of n.[1] Among the smallest such numbers are 2, with d(2) = 2 > d(1), and 4, with d(4) = 3 > d(2) and d(3) = 2.[1]Ramanujan's study of these numbers stemmed from their exceptional abundance of factors, highlighting the peaks in the growth of d(n).[1]
Role of the Divisor Function
The divisor function, commonly denoted d(n) or \tau(n), quantifies the number of positive divisors of a positive integer n.[2] For n with prime factorization n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}, where the p_i are distinct primes and the a_i are positive integers, d(n) is given by the formulad(n) = (a_1 + 1)(a_2 + 1) \cdots (a_k + 1).This multiplicative formula arises because each divisor of n is uniquely formed by choosing exponents from 0 to a_i for each prime p_i, yielding a_i + 1 choices per prime.[2][3]The function d(n) is multiplicative, meaning that if m and n are coprime (i.e., \gcd(m, n) = 1), then d(mn) = d(m) \cdot d(n).[3] This property stems from the unique factorization theorem in the integers and directly follows from the formula for d(n), as the prime factors of m and n are disjoint.[2] Multiplicativity simplifies computations for numbers with known factorizations and underpins many analytic properties of d(n).In the study of highly composite numbers, d(n) plays a pivotal role by serving as the primary metric for the "superiority" condition: a number n is highly composite if d(n) > d(m) for every positive integer m < n.[1] This record-setting behavior of d(n) distinguishes highly composite numbers from others, as verifying the status requires comparing d(n) against all smaller values, often through systematic enumeration or structural constraints.[4]To evaluate d(n) for verifying highly composite status, one first determines the prime factorization of n, which can be computationally intensive for large n but feasible using trial division or advanced factorization algorithms for candidates up to known bounds.[2] Once factored, the formula is applied directly to compute the product of the incremented exponents, providing an exact count without enumerating all divisors explicitly.[3]The growth of d(n) favors numbers with many small prime factors and nonincreasing exponents because the formula maximizes the product (a_i + 1) while keeping n = \prod p_i^{a_i} as small as possible relative to that product; smaller primes allow more factors or higher exponents before n exceeds a given size, thereby increasing d(n) more efficiently than using larger primes.[1] This structural preference explains why highly composite numbers tend to incorporate the smallest primes with exponents that decrease gradually, optimizing the divisor count for their magnitude.[4]
Historical Development
Ramanujan's Contributions
In 1915, Srinivasa Ramanujan introduced the concept of highly composite numbers in his seminal paper published in the Proceedings of the London Mathematical Society. He defined a highly composite number as a positive integer n that possesses more positive divisors than any smaller positive integer m < n, emphasizing its role in understanding the growth of the divisor function d(n). This work was motivated by Ramanujan's investigations into the divisor problem and the asymptotic behavior of d(n), building upon earlier results by mathematicians such as Dirichlet, Voronoi, and Landau on the maximum order of the number of divisors. His broader research on partition functions and related analytic number theory problems provided the context for exploring numbers that maximize divisor counts relative to their size.[1]Ramanujan observed that highly composite numbers typically exhibit a prime factorization of the form n = 2^{a_1} 3^{a_2} \cdots p_k^{a_k}, where the exponents are non-increasing (a_1 \geq a_2 \geq \cdots \geq a_k \geq 1) and resemble primorials (products of the first k primes) but with exponents adjusted to optimize the divisor count while keeping n as small as possible for a given d(n). Exceptions occur for small cases like 4 and 36, where the exponent sequence slightly deviates from strict monotonicity. In his paper, he compiled a comprehensive table listing the first 102 highly composite numbers, extending up to approximately $6.75 \times 10^{12} with d(n) = 10080, providing their prime factorizations and divisor counts to illustrate the pattern. Additionally, he presented a separate table of the first 50 superior highly composite numbers, a subclass where the ratio \frac{\log d(n)}{\log n} achieves local maxima, further highlighting structural properties.[1]Ramanujan conjectured that there are infinitely many highly composite numbers, reasoning from their asymptotic density and the continuous increase in the possible values of d(n) as n grows, though he did not provide a formal proof. This conjecture, supported by the observed progression in his tables, underscored the infinite nature of the sequence and invited further exploration into the distribution of such numbers. His contributions laid the foundational framework for subsequent studies in multiplicative number theory.[1]
Later Extensions and Research
In the 1940s, following Ramanujan's foundational work, mathematicians such as L. Alaoglu and Paul Erdős extended the theory of highly composite numbers by introducing related concepts like superabundant numbers and providing a precise characterization of their form. In their 1944 paper, they established that a number n is highly composite if and only if, for some r \geq 0, n maximizes the ratio d(m)/m^r among all positive integers m \leq n, where d(m) is the divisor function.[5] This formula offered a variational perspective, linking highly composite numbers to optimization problems in the divisor function and enabling deeper structural analysis.[6]Building on these ideas, Jean-Louis Nicolas advanced the study in the 1980s by exploring superior highly composite numbers—those introduced by Ramanujan as canonical forms achieving local maxima in the divisor density—and their limiting behavior. In his 2018 work, later elaborated in subsequent papers, Nicolas demonstrated that sequences of superior highly composite numbers converge to limits that refine bounds on the divisor function, providing asymptotic insights into their distribution.[7] Notably, Guy Robin, in his 1984 thesis, connected these numbers to the Riemann hypothesis through an equivalent criterion: the hypothesis holds if and only if the sum-of-divisors function \sigma(n)/n satisfies \sigma(n) < e^\gamma n \log \log n for all n > 5040, with near-equality cases occurring at superior highly composite numbers. Nicolas has since collaborated on extensions of this work.[8][9] This linkage underscores the role of highly composite numbers in major unsolved problems in analytic number theory.Computational efforts have significantly expanded the known repertoire of highly composite numbers since the mid-20th century. In the 1990s, Achim Flammenkamp computed extensive lists, starting with the first 1,200 terms and extending to over 779,000 by the early 2000s, encompassing numbers with thousands of digits and confirming their primality factors through rigorous verification.[10] These records, hosted in resources like the Online Encyclopedia of Integer Sequences (OEIS A002182), have facilitated empirical studies of patterns and growth.[11] Post-2020 research has built on this foundation, with algorithms for generalized superior highly composite numbers (for divisor powers k \geq 2) computing explicit bounds up to k=100, aiding explorations in computational number theory tools like SageMath and PARI/GP.[12]The structure of highly composite numbers also intersects with unsolved questions in prime distribution, as their prime factorizations require exponents that decrease with prime size, influenced by the density and gaps in primes. For instance, irregularities in prime spacing can constrain possible exponent sequences, tying the precise forms of these numbers to conjectures like the Riemann hypothesis or refined prime number theorems, though no full resolution exists.[7] Recent analyses, such as a 2023 study proving only finitely many integers n where both n and d(n) are highly composite, highlight ongoing theoretical refinements amid these open challenges.[13]
Structural Characterization
Prime Factorization Requirements
A highly composite number n must possess a prime factorization of the specific form n = 2^{a_1} 3^{a_2} 5^{a_3} \cdots p_k^{a_k}, where the exponents satisfy a_1 \geq a_2 \geq \cdots \geq a_k \geq 1 and the primes p_1 = 2 < p_2 = 3 < \cdots < p_k are the first k consecutive primes starting from 2.[1] This structure ensures that n has more divisors than any smaller positive integer, as the divisor function d(n) = (a_1 + 1)(a_2 + 1) \cdots (a_k + 1) is maximized relative to the magnitude of n.[1][14]The non-increasing sequence of exponents is essential because any deviation, such as increasing an exponent for a larger prime while decreasing one for a smaller prime, would yield a number m < n with d(m) \geq d(n), violating the highly composite property.[1] Similarly, omitting any intermediate prime (creating a gap in the sequence) or including a non-consecutive prime would allow for a smaller multiple with at least as many divisors, again undermining the superiority of d(n).[14] These constraints arise from the need to balance the growth of n against the multiplicative increase in d(n), where smaller primes contribute more efficiently to divisor count when assigned higher exponents.[1]This characterization of the prime factorization was first observed empirically by Srinivasa Ramanujan in his 1915 paper, where he tabulated numerous examples and inferred the pattern from their divisor-maximizing behavior.[1] Subsequent rigorous proofs, such as those establishing the necessity of consecutive primes and non-increasing exponents via lemmas on exponent bounds and prime gaps, formalized these requirements in the mid-20th century.[14]
Collatz-Wielandt Characterization
The characterization of highly composite numbers with the required prime factorization form relies on conditions ensuring that no smaller number has as many or more divisors. Alaoglu and Erdős (1944) proved that if n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} with consecutive primes p_1 < p_2 < \cdots < p_k and non-increasing exponents a_1 \geq a_2 \geq \cdots \geq a_k \geq 1, then n is highly composite provided the exponents satisfy specific optimality conditions derived from the divisor maximization principle. In particular, the exponent of the largest prime p_k is 1, except for the cases n=4=2^2 and n=36=2^2 3^2.[5]These conditions arise from analyzing perturbations to the factorization: reducing an exponent on a smaller prime and increasing one on a larger prime (or adding a new prime) must not yield a smaller m with d(m) ≥ d(n). The paper establishes asymptotic bounds on the exponents; for instance, if q is a prime factor with exponent k, and q is not too large relative to the largest prime p, then k \approx \frac{\log p}{\log q} \cdot \frac{\log \log p}{\log \log q} (more precisely, from Theorem 12, under suitable conditions, k = \frac{\log p \cdot \log \log p}{\log q \cdot \log \log q} [1 + O(1/(\log p)^c)]). Additionally, groups of equal exponents are limited, and exponents drop to 1 and remain 1 for the trailing primes.[5]A key result (Theorem 11) provides that if q is the greatest prime with exponent k >1, and under growth assumptions, \log(1 + 1/k) = \frac{\log q \cdot \log 2}{\log p} + O(1/q^{1-\theta} \log p), ensuring the structure prevents divisor count superiority by smaller numbers. Highly composite numbers correspond to the limiting case of superior highly composite numbers as the parameter ε → 0^+, where superior highly composite numbers maximize d(m)/m^ε for fixed ε >0. Examples like 12 = 2^2 · 3^1 and 60 = 2^2 · 3^1 · 5^1 satisfy these structural properties, with their exponent sequences verified to be optimal against smaller candidates.[5]
Examples and Illustrations
Small Highly Composite Numbers
The smallest highly composite numbers serve as foundational examples, demonstrating how the divisor function d(n) achieves new maxima at each step. These numbers typically feature prime factorizations where the exponents are non-increasing, incorporating the smallest primes with optimized powers to maximize the number of divisors relative to their size.The first 30 highly composite numbers, along with their prime factorizations and divisor counts, are presented in the table below. This enumeration originates from the systematic study by Ramanujan, who compiled extensive tables of such numbers.[15]
Order
n
Prime Factorization
d(n)
1
1
(none)
1
2
2
$2^1
2
3
4
$2^2
3
4
6
$2^1 \cdot 3^1
4
5
12
$2^2 \cdot 3^1
6
6
24
$2^3 \cdot 3^1
8
7
36
$2^2 \cdot 3^2
9
8
48
$2^4 \cdot 3^1
10
9
60
$2^2 \cdot 3^1 \cdot 5^1
12
10
120
$2^3 \cdot 3^1 \cdot 5^1
16
11
180
$2^2 \cdot 3^2 \cdot 5^1
18
12
240
$2^4 \cdot 3^1 \cdot 5^1
20
13
360
$2^3 \cdot 3^2 \cdot 5^1
24
14
720
$2^4 \cdot 3^2 \cdot 5^1
30
15
840
$2^3 \cdot 3^1 \cdot 5^1 \cdot 7^1
32
16
1260
$2^2 \cdot 3^2 \cdot 5^1 \cdot 7^1
36
17
1680
$2^4 \cdot 3^1 \cdot 5^1 \cdot 7^1
40
18
2520
$2^3 \cdot 3^2 \cdot 5^1 \cdot 7^1
48
19
5040
$2^4 \cdot 3^2 \cdot 5^1 \cdot 7^1
60
20
7560
$2^3 \cdot 3^3 \cdot 5^1 \cdot 7^1
64
21
10080
$2^5 \cdot 3^2 \cdot 5^1 \cdot 7^1
72
22
15120
$2^4 \cdot 3^3 \cdot 5^1 \cdot 7^1
80
23
20160
$2^6 \cdot 3^2 \cdot 5^1 \cdot 7^1
84
24
25200
$2^4 \cdot 3^2 \cdot 5^2 \cdot 7^1
90
25
27720
$2^3 \cdot 3^2 \cdot 5^1 \cdot 7^1 \cdot 11^1
96
26
45360
$2^4 \cdot 3^4 \cdot 5^1 \cdot 7^1
100
27
50400
$2^5 \cdot 3^2 \cdot 5^2 \cdot 7^1
108
28
55440
$2^4 \cdot 3^2 \cdot 5^1 \cdot 7^1 \cdot 11^1
120
29
83160
$2^3 \cdot 3^3 \cdot 5^1 \cdot 7^1 \cdot 11^1
128
30
110880
$2^5 \cdot 3^2 \cdot 5^1 \cdot 7^1 \cdot 11^1
144
Among these, the factorials from 1! to 7! (1, 2, 6, 24, 120, 720, 5040) are all highly composite, as are the primorials 2 and 6 (the products of the first one and two primes, respectively).[15] By definition, each highly composite number n satisfies d(n) > d(m) for all positive integers m < n, which is verified by the strictly increasing sequence of divisor counts in the table above.[15]
Patterns and Generation Methods
Highly composite numbers display characteristic patterns in their prime factorizations, where the exponents of successive primes are nonincreasing: if n = 2^{a_1} 3^{a_2} \cdots p_k^{a_k}, then a_1 \geq a_2 \geq \cdots \geq a_k \geq 1, with the primes being the initial segment of primes up to p_k.[1] For sufficiently large such numbers, the exponents decrease strictly in the initial part, forming groups of equal values only toward the end, such as sequences of 1s or 2s.[1]These exponent patterns often approximate those of superior highly composite numbers, where the exponents follow a_p = \left\lfloor \frac{c}{\log p} \right\rfloor for some constant c > 0; highly composite numbers arise as integer scalings of these forms that preserve the divisor-maximizing property.[1] Ramanujan noted that the early exponents satisfy relations like a_\lambda \log \lambda \sim \frac{\log p}{\log 2}, where p is the largest prime factor, for indices \lambda, reflecting a logarithmic decay that ensures the divisor count grows optimally.[1]A practical method to generate highly composite numbers begins with 1 and proceeds iteratively: from the current highly composite number h, identify the smallest integer greater than h that has more divisors than h and all intermediates, typically obtained by multiplying h by a small prime power while keeping exponents nonincreasing.[4] This approach was systematized by Siano, who split h into a stable prefix (product of initial primes to power 1) and a variable suffix (higher powers of smaller primes), then generate and test candidate suffixes within a bounded range (e.g., between h and $2h) to find the next candidate with increased divisor count.[16]Computational implementations of such algorithms have extended Ramanujan's manual list of the first 102 highly composite numbers up to one exceeding $10^{12} with d(N) = 10{,}080 to over $10^{18}, with challenges arising from the exponential growth in exponents for small primes like 2 and 3, requiring efficient divisor computation and factorization checks for large candidates.[17] For instance, starting from small examples like 12 = $2^2 \times 3^1, the method yields 24 = $2^3 \times 3^1 as the next by multiplying by 2, preserving the pattern.[4]
Key Properties
Divisor Count Superiority
Highly composite numbers exhibit a defining property of divisor count superiority: for any such number n, the divisor function d(n), which counts the positive divisors of n, satisfies d(n) > d(m) for all positive integers m < n. This means each highly composite number sets a new record for the maximum number of divisors achieved by any integer up to that point.[4][11]The proof of this superiority stems from the specific prime factorization form of highly composite numbers, where n = 2^{a_1} 3^{a_2} \cdots p_k^{a_k} with primes p_i in increasing order and exponents a_1 \geq a_2 \geq \cdots \geq a_k \geq 1. The divisor count is then d(n) = \prod_{i=1}^k (a_i + 1). If this form is deviated from—such as by increasing an exponent on a larger prime while decreasing one on a smaller prime—a smaller integer m < n can be constructed with d(m) \geq d(n), contradicting the maximality. For instance, replacing a higher power of a large prime with additional factors of smaller primes yields a number with the same or greater divisor product but reduced magnitude. This structural requirement ensures the exponents are optimally decreasing to maximize d(n)/\log n, though the focus here is on the absolute record-setting behavior.[1]As a consequence, highly composite numbers form a strictly increasing chain in terms of their divisor counts, where each subsequent number h_{k+1} is the smallest integer achieving at least d(h_k) + 1 divisors, thereby extending the record without gaps in the sequence of maximal values. This chain-like progression underscores their role as milestones in the divisor function's growth.[4][1]Compared to typical positive integers, the divisor count d(n) for highly composite numbers grows notably faster; while the average d(m) for m \leq n remains close to \log n or slower, highly composite numbers achieve divisor counts that outpace this by orders of magnitude even at modest sizes, such as d(60) = 12 versus an average around 3 for numbers up to 60.[1][11]Finally, this record-setting property implies uniqueness: no two highly composite numbers share the same divisor count, as each establishes a strictly larger d(n) than all predecessors, including prior highly composite numbers.[11][4]
Multiplicativity and Closure
The set of highly composite numbers (HCN) is multiplicative in the sense that their prime factorizations follow a specific form with non-increasing exponents, but the set itself is not closed under multiplication.[1] That is, the product of two HCN is not necessarily itself an HCN, though it may be if the resulting exponents maintain the required decreasing order and the divisor function value exceeds that of all smaller integers.[18] For instance, $4 \times 6 = 24, where both 4 and 6 are HCN and 24 is also an HCN, as its 8 divisors surpass those of all smaller numbers.[18] In contrast, $12 \times 12 = 144 yields a number with 15 divisors, fewer than the 16 divisors of 120 (which is smaller than 144 and an HCN), so 144 is not highly composite.[18]A partial closure property holds when multiplying an HCN by a prime power: such a product may result in another HCN if it preserves the non-increasing exponent sequence in the prime factorization and increases the divisor count appropriately relative to intervening numbers.[1] For example, starting from the HCN 12 ($2^2 \times 3), multiplying by the prime power $2^1 gives 24 ($2^3 \times 3), which is HCN, and multiplying by $3^1 gives 36 ($2^2 \times 3^2), also HCN.[18] This generative process underlies algorithms for producing successive HCN, where each new one is obtained by multiplying a prior HCN by a small rational factor (often a prime power) up to 2 while verifying the divisor condition.[18]Regarding semigroup structure, the set of HCN under multiplication does not form a semigroup, as it lacks closure—the product of elements need not remain in the set, and the ratios of consecutive HCN approach 1, implying dense distribution that disrupts consistent maximization of divisors in products.[1] However, the HCN can be viewed as generators in the broader multiplicative semigroup of positive integers with non-increasing prime exponents, where successive HCN are built iteratively from prior ones via compatible prime power multiplications.[18]
Growth and Distribution
Asymptotic Growth Bounds
The asymptotic growth of highly composite numbers and their divisor counts is closely tied to the maximal order of the divisor function d(n), as these numbers realize the record values of d(n) up to their magnitude. Ramanujan established early bounds assuming the prime number theorem, showing that \log d(n) \sim \mathrm{Li}(\log n) for the maximal order, where \mathrm{Li}(x) = \int_0^x \frac{dt}{\log t} is the logarithmic integral function; more precisely, d(n) < \exp\left( c \frac{\log n}{\log \log n} \right) for some constant c > 0 and all sufficiently large n. This implies an upper bound of the form d(n) = O\left( \exp\left( 2 \frac{\log n}{\log \log n} + O\left( \frac{\log n}{(\log \log n)^2} \right) \right) \right), with a matching lower bound achieved for infinitely many n.[1]Wigert refined this to the precise maximal order, demonstrating that the divisor function satisfies\limsup_{n \to \infty} \frac{\log d(n) \cdot \log \log n}{\log n} = \log 2.Equivalently, the maximal order is \exp\left( (\log 2 + o(1)) \frac{\log n}{\log \log n} \right), meaning d(n) < \exp\left( \log 2 \cdot \frac{\log n}{\log \log n} (1 + o(1)) \right) for all n, with equality in the limit superior achieved along a subsequence.[19] This refinement sharpens Ramanujan's estimate by determining the optimal constant \log 2 \approx 0.693.For the sequence of highly composite numbers n_k, where n_1 = 1 < n_2 = 2 < \cdots and d(n_k) > d(m) for all m < n_k, the growth follows a similar pattern but with explicit structural ties to primes. Let p_k denote the largest prime factor of n_k. Then \log n_k \sim \theta(p_k), where \theta(x) = \sum_{p \leq x} \log p is the Chebyshev function; since \theta(x) \sim x by the prime number theorem, this yields \log n_k \sim p_k. Consequently, the divisor count satisfies d(n_k) \sim \exp\left( c \frac{\log n_k}{\log \log n_k} \right) for some c > 0, aligning with the general maximal order and reflecting the optimized prime factorization of highly composite numbers, where exponents decrease monotonically and incorporate all primes up to p_k.[1][12]
Density in the Integers
The set of highly composite numbers possesses natural density zero among the positive integers. This follows from the fact that the counting function Q(x), representing the number of highly composite numbers not exceeding x, grows much more slowly than x, specifically Q(x) = o(x) as x \to \infty. The underlying reason is the super-exponential growth of the sequence of highly composite numbers, where the k-th term n_k satisfies \log n_k \sim k^{1/a} for some a > 0, implying that Q(x) \sim (\log x)^a up to constants.[5]Alaoglu and Erdős established a lower bound for the counting function, proving that Q(x) > (\log x)^{1+c} for some constant c > 0 and all sufficiently large x. An upper bound of the form Q(x) < (\log x)^a \log \log x for some a > 0 also holds, confirming that Q(x) is polylogarithmic in x. These bounds underscore the extreme rarity of highly composite numbers, as even the lower estimate places them far sparser than primes, whose count is asymptotic to x / \log x.[5]The logarithmic density of highly composite numbers is likewise zero. This is because the series \sum 1/n over all highly composite n converges, due to the rapid growth of n_k, so the partial sums up to x remain bounded as x \to \infty, and dividing by \log x yields zero. Nonetheless, highly composite numbers can be viewed as "dense" within the subset of integers possessing a superior number of divisors, as they exclusively achieve the successive maxima of the divisor function d(n), with superior highly composite numbers providing the optimal realizations for normalized variants like d(n)/n^\epsilon.[5][1]As a consequence of their growth, the gaps between consecutive highly composite numbers n_k and n_{k+1} increase rapidly. Ramanujan showed that the ratio n_{k+1}/n_k \to 1 as k \to \infty, but the additive gaps satisfy n_{k+1} - n_k = O\left( n_k \frac{(\log \log \log n_k)^{3/2}}{\sqrt{\log \log n_k}} \right), which grows without bound and faster than any fixed multiple of smaller terms, reflecting the accelerating sparsity.[1]
Related Concepts
Superior Highly Composite Numbers
Superior highly composite numbers generalize the concept of highly composite numbers by considering a weighted version of the divisor function. A positive integer n is a superior highly composite number if there exists some \varepsilon > 0 such that \frac{d(m)}{m^\varepsilon} < \frac{d(n)}{n^\varepsilon} for all positive integers m < n. This condition identifies numbers where the ratio of the number of divisors to a power of the number achieves a strict local maximum relative to smaller integers.[1]Ramanujan observed that every superior highly composite number is also highly composite, since the condition for \varepsilon > 0 implies the unweighted divisor count superiority when \varepsilon is sufficiently small, but the converse does not hold: there exist highly composite numbers that fail the stricter weighted criterion for any \varepsilon > 0. Highly composite numbers can be viewed as arising from scaling or integer-part approximations of superior highly composite structures, particularly in their prime factorizations.[1]In their prime factorization n = \prod_p p^{a_p}, superior highly composite numbers exhibit exponents a_p = \lfloor c / \log p \rfloor for some constant c > 0, with the exponents nonincreasing across successive primes and the primes taken consecutively from 2 up to some largest prime depending on c. This form ensures the exponents decrease smoothly, reflecting the optimization of the divisor-to-size ratio under the weighted measure.[1]These numbers play a key role in bounding the maximal order of the divisor function d(n). Specifically, the superior highly composite numbers attain the maximum value of d(n)/n^\varepsilon for each \varepsilon > 0, enabling precise upper bounds such as d(n) < n^\varepsilon \exp\left( C \frac{\log n}{\log \log n} \right) for suitable constants C, derived from the structure of these numbers.[1]
Connections to Other Sequences
Highly composite numbers exhibit strong connections to several other important sequences in number theory, particularly those involving abundance of divisors or specific forms of factorization.All highly composite numbers greater than 12 are abundant, meaning the sum of their proper divisors exceeds the number itself (σ(n) > 2n, where σ(n) is the sum-of-divisors function). This property arises from their structure, which ensures a high divisor count relative to their magnitude, leading to abundant divisor sums for larger instances. For example, 24 has σ(24) = 60 > 48, and the pattern holds beyond this point due to the non-decreasing exponents in their prime factorizations that favor abundance.[4][5]Colossally abundant numbers form a related sequence where the numbers maximize σ(n)/n^ε for some ε > 0 and all smaller n, with exponents in the prime factorization determined by logarithmic rules such as k_q ≈ log((q^{1+ε} - 1)/(q^ε - 1)) / log q. These numbers represent a subset intersecting with highly composite numbers, as the first 15 colossally abundant numbers coincide with the first 15 superior highly composite numbers, which are themselves a special case of highly composite numbers with exponents strictly decreasing and following similar log-based bounds. This overlap highlights how highly composite numbers can achieve extremal divisor properties under refined abundance criteria.[20]Every primorial (the product of the first k primes, sequence A002110 in the OEIS) is a highly composite number, as their square-free structure with consecutive small primes maximizes the divisor count up to that point. More broadly, every highly composite number (except 1) can be expressed as a finite product of primorials, reflecting the non-increasing exponent pattern in their prime factorizations that builds upon primorial bases. For instance, 60 = 2 × 30, where both are primorials.[11][21]Many factorials are also highly composite numbers, specifically the first seven: 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, 6! = 720, and 7! = 5040, each setting a record for the number of divisors among smaller integers. However, 8! = 40320 is not highly composite, as it has the same number of divisors (96) as 27720, which is smaller, so it does not have more divisors than all smaller integers. This connection underscores the role of highly composite numbers in factorial-like products.[11][22]The sequence of highly composite numbers is cataloged as A002182 in the Online Encyclopedia of Integer Sequences (OEIS), where it links to related sequences such as A000142 (factorials), A002110 (primorials), and A004394 (superabundant numbers, which share the first 19 terms with highly composite numbers). These interconnections reveal highly composite numbers as a foundational sequence bridging divisor records with other extremal forms in multiplicative number theory.[11]