Superior highly composite number
A superior highly composite number is a positive integer n for which there exists some \epsilon > 0 such that \frac{d(n)}{n^\epsilon} \geq \frac{d(k)}{k^\epsilon} for all positive integers k, where d(m) denotes the number of positive divisors of m.[1][2] These numbers represent the maxima of the divisor function normalized by a power of the integer itself, providing a refined class within highly composite numbers, which are positive integers with strictly more divisors than any smaller positive integer.[2][3] Introduced by the Indian mathematician Srinivasa Ramanujan in his 1915 paper "Highly Composite Numbers," superior highly composite numbers extend the study of numbers with unusually many divisors, motivated by their role in asymptotic estimates for d(n).[1] Ramanujan proved that every such number has a prime factorization of the form n = 2^{a_1} 3^{a_2} 5^{a_3} \cdots p_r^{a_r}, where the exponents a_1 \geq a_2 \geq \cdots \geq a_r \geq 1 are non-increasing, the primes p_i are the first r primes, and the exponents satisfy specific inequalities derived from logarithmic considerations.[1][2] Key properties include that all superior highly composite numbers are also highly composite, but the converse does not hold, and they achieve equality in certain upper bounds for the divisor function, such as d(n) < C_\epsilon n^\epsilon for any \epsilon > 0 and suitable constant C_\epsilon.[1] The sequence begins with 2, 6, 12, 60, 120, 360, 2520, 5040, and continues infinitely, listed in OEIS A002201.[3] These numbers have applications in analytic number theory, particularly in deriving explicit bounds for the growth of d(n) and in problems related to the distribution of prime factors.[4]Definition and Motivation
Formal Definition
A positive integer n is a superior highly composite number if there exists a positive real number \epsilon > 0 such that \frac{d(n)}{n^{\epsilon}} \geq \frac{d(m)}{m^{\epsilon}} for all positive integers m, where d(k) denotes the number of positive divisors of k.[1] Ramanujan's original formulation specified \geq for m < n and strict > for m > n, which is equivalent to the modern global maximization in practice.[2] This inequality characterizes n as achieving the maximum value of the adjusted divisor count d(k)/k^{\epsilon} over all k, for some exponent \epsilon > 0. The ratio d(n)/n^{\epsilon} serves as a measure of divisor density, emphasizing numbers with exceptionally many divisors relative to their size, scaled by the power \epsilon to capture "superior" performance across a range of sensitivities to growth.[1] Superior highly composite numbers constitute a proper subset of the highly composite numbers, inheriting their property of having more divisors than any smaller positive integer while satisfying the stronger uniformity condition via \epsilon.[1] 1 is sometimes considered the smallest superior highly composite number (as in OEIS A002201), though excluded in some definitions including Ramanujan's original due to lack of prime factors. For appropriate large \epsilon > 1, d(1)/1^{\epsilon} = 1 > d(m)/m^{\epsilon} for all m > 1, since d(m) < m^{\epsilon}.[3][1]Historical Motivation
The historical motivation for superior highly composite numbers originates from the quest to characterize natural numbers that possess an exceptionally large number of divisors relative to their magnitude, a key problem in analytic number theory for understanding the extremal behavior of the divisor function d(n). Prior work had established the average order of d(n), as explored by Dirichlet, Voronoi, and Landau, but the maximum or irregular growth remained less understood, with Wigert providing early upper bounds such as d(n) < \exp\left( C \frac{\log n}{\log \log n} \right) for some constant C.[1][5] Ramanujan sought to identify the numbers achieving these upper bounds in a precise manner, recognizing that such extremal values occur at specific discrete points rather than continuously.[1] This pursuit connects to broader concepts of numerical abundance, where superior highly composite numbers maximize d(n)/n^\epsilon for some \epsilon > 0. These maximizations prove useful in analytic number theory for approximating the growth of divisor-related functions and bounding arithmetic functions in problems involving prime factorizations.[1][4] Ramanujan's framework thus provides tools for such theoretical bounds.[1] Intuitively, early interest in such "superior" numbers arose from observations of small integers like 12 (with 6 divisors) and 60 (with 12 divisors), which outperform smaller numbers in divisor count without being excessively large, hinting at an underlying pattern in highly divisible structures. Ramanujan's key insight was that superior highly composite numbers form a countable sequence with explicit prime factorization rules, enabling rigorous analysis of their role in achieving the normal maximum order of d(n). This structure not only elucidates theoretical bounds but also ties into deeper conjectures, such as the Riemann hypothesis, under which these numbers precisely realize the upper limits for d(n).[1]Historical Development
Ramanujan's Original Work
Srinivasa Ramanujan introduced superior highly composite numbers in his 1915 paper "Highly Composite Numbers," published in the Proceedings of the London Mathematical Society, where he refined the earlier concept of highly composite numbers by focusing on a stricter subclass distinguished by their maximal value of the divisor function normalized by a power of the number relative to all other positive integers.[1] Ramanujan's approach centered on positive integers n for which there exists some \epsilon > 0 such that \frac{d(n)}{n^\epsilon} \geq \frac{d(k)}{k^\epsilon} for all positive integers k, where d(m) is the number of positive divisors of m. This ensures these numbers achieve an optimal divisor density. He constructed such numbers explicitly through their prime factorization, requiring the exponents to form a non-increasing sequence.[1] In a key theorem, Ramanujan proved that every superior highly composite number takes the form p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}, where p_1 < p_2 < \cdots < p_k are the first k primes and the exponents satisfy a_1 \geq a_2 \geq \cdots \geq a_k \geq 1. This structural condition guarantees the superior property by maximizing the divisor function under the given constraints.[1] Ramanujan provided a table of superior highly composite numbers, with the first few being 2, 6, 12, 60, 120, 360, 840, 1260, 2520, and 5040, illustrating the progressive inclusion of smaller primes with diminishing exponents to sustain the superior property. These initial results laid the groundwork for understanding the discrete nature of numbers optimizing divisor functions.[1]Post-Ramanujan Advancements
In the 1940s, Leonidas Alaoglu and Paul Erdős provided rigorous proofs confirming several of Ramanujan's conjectures regarding superior highly composite numbers (SHCNs), including their precise canonical form in terms of decreasing exponents in the prime factorization and the infinitude of such numbers.[6] Their work extended the theory by characterizing SHCNs as a subclass of highly composite numbers where the ratio of the divisor function to the number raised to any positive power exceeds that of all larger integers, establishing foundational results on their structure.[7] During the 1980s, Jean-Louis Nicolas advanced the study by linking SHCNs to extremal problems in the divisor function and refinements related to the prime number theorem. Concurrently, Guy Robin's 1984 work on inequalities for the sum-of-divisors function \sigma(n) connected colossally abundant numbers—a class analogous to SHCNs but optimizing \sigma(n)/n—to the Riemann hypothesis, showing that \sigma(n) < e^\gamma \log \log n + 0.6483\ldots / \log \log n for all n \geq 5041 if and only if the hypothesis holds, with colossally abundant numbers marking the points where this bound is nearly tight. While SHCNs focus on d(n), these developments highlight parallels in numbers with many divisors or sums of divisors. Computational progress in the 1990s, led by Achim Flammenkamp, extended the enumeration of highly composite numbers—which include SHCNs as a subset—to over 700,000 terms using optimized algorithms that systematically check exponent conditions for primorial-like structures.[8] These efforts facilitated the identification of larger SHCNs by leveraging computer-assisted verification of divisor maximality. Recent developments up to 2025 have integrated SHCN generation with computer algebra systems like SageMath, enabling efficient computation of terms with thousands of digits and exploration of their properties in broader contexts.[4] For instance, a 2025 study utilized SHCNs to derive explicit upper bounds for generalized divisor functions, highlighting their utility in analytic number theory.[9] Ongoing research also connects SHCNs to unsolved problems, such as the precise density of their distribution among integers, remaining an open challenge in additive number theory.[10]Mathematical Properties
Abundancy and Divisor Function
The abundancy index of a positive integer n is the ratio \frac{\sigma(n)}{n}, where \sigma(n) is the sum-of-divisors function, which counts the sum of all positive divisors of n. For a superior highly composite number n with prime factorization n = \prod p_i^{a_i}, the multiplicativity of \sigma yields the explicit formula \frac{\sigma(n)}{n} = \prod_i \frac{1 - p_i^{-(a_i + 1)}}{1 - p_i^{-1}}, where the product is over the distinct primes p_i dividing n, and the exponents a_i are non-increasing with the primes p_i as dictated by the superiority condition. This formula arises from the geometric series summation for each prime power: \sigma(p_i^{a_i}) = 1 + p_i + \cdots + p_i^{a_i} = \frac{p_i^{a_i + 1} - 1}{p_i - 1}, so \frac{\sigma(p_i^{a_i})}{p_i^{a_i}} = \frac{1 - p_i^{-(a_i + 1)}}{1 - p_i^{-1}}. The decreasing exponents in superior highly composite numbers ensure that the product is optimized by favoring small primes with higher powers, leading to a relatively large abundancy index compared to numbers of similar magnitude.[1] The superiority condition on the exponents a_i, which requires that there exists \epsilon > 0 such that \frac{d(n)}{n^\epsilon} \geq \frac{d(k)}{k^\epsilon} for all positive integers k, indirectly constrains the abundancy index to achieve values near the maximal possible order. Specifically, for superior highly composite n, \frac{\sigma(n)}{n} > \log \log n + B for some constant B > 0. More precisely, the maximal order is given by \limsup_{n \to \infty} \frac{\sigma(n)/n}{\log \log n} = e^\gamma, where \gamma \approx 0.57721 is the Euler-Mascheroni constant; while colossally abundant numbers approach this bound, superior highly composite numbers also exhibit high abundancy due to their factorization structure.[1][6] Superior highly composite numbers achieve local maxima in abundancy because their exponent configuration balances the trade-off between increasing the sum of divisors and the growth in n. To see this, consider modifying n by either increasing an exponent a_j for some prime p_j or multiplying by a new prime q > all p_i. Increasing a_j to a_j + 1 multiplies the abundancy by \frac{1 + 1/p_j^{a_j + 1}}{1 + 1/p_j^{a_j}}, which is greater than 1 but approaches 1 as a_j grows large, while n increases by a factor of p_j; for large enough a_j, this modification decreases the overall \frac{\sigma(n')}{n'}. Similarly, multiplying by a new prime q multiplies the abundancy by $1 + 1/q < 2, but n increases by q; if q is sufficiently large, $1 + 1/q < 1 + \log q / q in logarithmic terms, leading to a net decrease in the ratio. The superiority condition ensures the exponents are set such that any such modification would violate the maximality for the divisor count ratio d(n)/n^\epsilon, and the analogous logarithmic comparison for \sigma(n)/n (replacing d with \sigma in the optimization) shows that the configuration also locally maximizes the abundancy, with the ratio decreasing upon further modifications until the next superior highly composite number.[6]Multiplicative and Order Properties
Superior highly composite numbers exhibit a strictly increasing order in their sequence, commencing with 1 (sometimes excluded) and proceeding as 2, 6, 12, 60, 120, 360, 2520, 5040, and continuing onward. Each successive number is generated by multiplying the prior term by a prime or a prime power, ensuring the resulting number maximizes the ratio of the divisor function to a power of the number for some positive exponent e. This construction guarantees that the sequence is ordered by magnitude, with no overlaps or repetitions.[1] The set of superior highly composite numbers is not closed under multiplication, meaning the product of two such numbers is generally not superior highly composite. For example, both 6 and 12 are superior highly composite, but 6 × 12 = 72 is not, as 72 fails to achieve the required maximization of d(n)/n^e for any e > 0. However, products of specific superior highly composite numbers can yield another under conditions where the multiplication aligns with the prescribed decreasing exponent pattern in the prime factorization, such as extending the factorization by adding a new prime factor with exponent 1 or increasing an existing exponent appropriately.[1] Each superior highly composite number possesses a unique prime factorization, with exponents that are non-increasing across the primes in a manner determined by a parameter related to e, ensuring no two share identical exponents for the same set of primes. This uniqueness stems from the parametric form introduced by Ramanujan, where each number corresponds to a distinct configuration of exponents optimized for a particular e.[1] Superior highly composite numbers have zero natural density in the natural numbers, as their count grows much slower than any positive power of x. The asymptotic number of such numbers up to x is on the order of \frac{\log x}{\log \log x}, reflecting the prime-counting nature of the parameters governing their construction. Although the natural density is zero, the set has positive logarithmic density due to the divergent sum of reciprocals over these numbers.[11][1]Prime Factorization Structure
Exponent Conditions
Superior highly composite numbers exhibit a specific structure in their prime factorization. Every such number n can be expressed as n = \prod_{i=1}^k p_i^{a_i}, where p_1 = 2 < p_2 = 3 < \cdots < p_k are the first k primes, and the exponents satisfy a_1 \geq a_2 \geq \cdots \geq a_k \geq 1. This non-increasing sequence of exponents ensures that the factorization prioritizes smaller primes with higher powers, optimizing the divisor-related properties central to the definition. This form is a fundamental characteristic derived from the maximization principles underlying these numbers.[1] The superiority condition imposes strict constraints on these exponents to maintain the number's optimal status. Specifically, for the largest prime p_k in the factorization, introducing a new prime p > p_k with exponent 1 (or higher) or increasing any existing exponent a_i must not produce a multiple m > n such that d(m)/m^\epsilon > d(n)/n^\epsilon for the \epsilon > 0 associated with n. This local maximality ensures that n achieves a peak in the divisor function normalized by a power relative to nearby multiples, preventing any simple adjustment from improving the ratio. Such conditions guarantee that superior highly composite numbers form a discrete sequence where each entry dominates in this metric over a range of scales.[1] Ramanujan's criterion provides a precise way to determine these exponents: for a parameter x > 0, the exponent of the prime p is given by a_p = \left\lfloor p^{1/x} - 1 \right\rfloor, with the primes taken as the first k such that p_k \leq 2^x and the sequence is non-increasing. This form arises from the iterative optimization process in constructing the sequence, where exponents are chosen to balance the contribution of each prime while preserving the non-increasing order and overall superiority. The criterion links the structure directly to the broader class of highly composite numbers, ensuring consistency across the hierarchy.[1] For instance, consider n = 12 = 2^2 \times 3^1, an early superior highly composite number. Here, the exponents 2 and 1 are non-increasing, and the configuration satisfies local maximality: multiplying by the next prime 5 to get 60 increases the size but aligns with the sequence's progression, while alternatives like $2^3 \times 3 = [24](/page/24) or $2^2 \times 3^2 = [36](/page/36) do not yield a superior ratio at that scale without violating the conditions. This example illustrates how the exponent rules enforce the number's position in the sequence.[1]Radices and Their Role
In the theory of superior highly composite numbers (SHCN), the sequence of exponents (a_1, a_2, \dots, a_k) in the prime factorization of an SHCN, where n = 2^{a_1} \cdot 3^{a_2} \cdot \dots \cdot p_k^{a_k} with a_1 \geq a_2 \geq \dots \geq a_k \geq 1 and p_k the k-th prime, encapsulates the structural form of the number.[1] This sequence ensures the exponents decrease or remain constant as larger primes are incorporated. The sequence of exponents plays a central role in defining and generating SHCN by determining the "shape" of their prime factorizations; each SHCN corresponds uniquely to one such sequence, and the superiority property requires that the sequence maximizes d(n)/n^\epsilon for some \epsilon > 0 relative to all smaller numbers.[1] Transitions between sequences of consecutive SHCN typically occur through two mechanisms: appending a new exponent a_{k+1} = 1 for the next prime to extend the sequence, or adjusting existing exponents while preserving the non-increasing order to optimize the divisor function growth. This constructive process ensures that SHCN form a chain where each subsequent sequence builds upon the previous to achieve ever-higher relative divisor counts. Ramanujan illustrated these sequences in his foundational table of SHCN, beginning with the trivial case of 1 corresponding to the empty sequence, followed by (1) for $2 = 2^1, (1,1) for $6 = 2^1 \cdot 3^1, and (2,1) for $12 = 2^2 \cdot 3^1, demonstrating the initial pattern of strict monotonic decrease before allowing plateaus in later entries.[1][3] These examples highlight how sequences evolve to incorporate more primes only when it enhances the overall divisor density, maintaining the non-increasing property throughout. A sequence of exponents is deemed "superior" precisely when it maximizes the divisor function ratio for its fixed length k, balancing the trade-off between including more small primes with higher exponents and adding larger primes with minimal exponents. Extending this to an infinite sequence, where the sequence continues indefinitely across all primes, yields a limiting form tied to the Euler product for the Riemann zeta function \zeta(s) = \prod_p (1 - p^{-s})^{-1}, as the optimal exponents align with the geometric series expansions that bound the maximal order of the divisor function.[1]Enumeration and Examples
Generation Methods
Superior highly composite numbers can be generated using methods based on their prime factorization structure, as described by Ramanujan. Each such number corresponds to a real parameter x > 0, with the form n = \prod_{i=1}^r p_i^{a_i}, where p_i are the first r primes, the exponents a_i are non-increasing positive integers given by a_i = \left\lfloor p_i^{1/x} - 1 \right\rfloor (or adjusted to satisfy boundary conditions), and r is chosen such that p_r \leq 2^x < p_{r+1}. By decreasing x from infinity (yielding n=1) in suitable steps, one obtains all superior highly composite numbers in increasing order. This parametric approach ensures the superiority condition by deriving exponents from logarithmic inequalities that maximize \log d(n) - \epsilon \log n.[1] Computational implementations adapt this by enumerating possible non-increasing exponent sequences over the first primes, verifying the superiority via bounds on the divisor function or direct computation of d(n). Software like SageMath and PARI/GP supports arbitrary-precision calculations for factorizations and divisor counts, enabling generation of terms up to very large sizes. For instance, algorithms have computed superior highly composite numbers with hundreds of prime factors. Verification for candidates involves checking that no smaller number achieves a higher d(n)/n^\epsilon for the associated \epsilon, often using theoretical estimates on the growth of d(n) to avoid exhaustive searches.[12][13][14]List of Small Superior Highly Composite Numbers
The superior highly composite numbers form a sequence that grows rapidly, with each term incorporating additional primes or increased exponents in a manner that maximizes the divisor density relative to their size. This list focuses on the first 20 terms (excluding 1, as per OEIS A002201), illustrating their prime factorizations and the number of divisors d(n), which increases with each term. These values highlight the rapid growth in the number of divisors characteristic of these numbers.[3][1]| Index | Number n | Prime Factorization | d(n) |
|---|---|---|---|
| 1 | 2 | $2^1 | 2 |
| 2 | 6 | $2^1 \times 3^1 | 4 |
| 3 | 12 | $2^2 \times 3^1 | 6 |
| 4 | 60 | $2^2 \times 3^1 \times 5^1 | 12 |
| 5 | 120 | $2^3 \times 3^1 \times 5^1 | 16 |
| 6 | 360 | $2^3 \times 3^2 \times 5^1 | 24 |
| 7 | 2520 | $2^3 \times 3^2 \times 5^1 \times 7^1 | 48 |
| 8 | 5040 | $2^4 \times 3^2 \times 5^1 \times 7^1 | 60 |
| 9 | 55440 | $2^4 \times 3^2 \times 5^1 \times 7^1 \times 11^1 | 100 |
| 10 | 720720 | $2^4 \times 3^2 \times 5^1 \times 7^1 \times 11^1 \times 13^1 | 120 |
| 11 | 1441440 | $2^5 \times 3^2 \times 5^1 \times 7^1 \times 11^1 \times 13^1 | 144 |
| 12 | 4324320 | $2^5 \times 3^3 \times 5^1 \times 7^1 \times 11^1 \times 13^1 | 192 |
| 13 | 21621600 | $2^5 \times 3^3 \times 5^2 \times 7^1 \times 11^1 \times 13^1 | 288 |
| 14 | 367567200 | $2^5 \times 3^3 \times 5^2 \times 7^1 \times 11^1 \times 13^1 \times 17^1 | 576 |
| 15 | 6983776800 | $2^6 \times 3^3 \times 5^2 \times 7^1 \times 11^1 \times 13^1 \times 17^1 | 768 |
| 16 | 13967553600 | $2^6 \times 3^4 \times 5^2 \times 7^1 \times 11^1 \times 13^1 \times 17^1 | 1296 |
| 17 | 321253732800 | $2^6 \times 3^4 \times 5^2 \times 7^2 \times 11^1 \times 13^1 \times 17^1 | 1728 |
| 18 | 2248776129600 | $2^7 \times 3^4 \times 5^2 \times 7^2 \times 11^1 \times 13^1 \times 17^1 | 2304 |
| 19 | 65214507758400 | $2^7 \times 3^4 \times 5^2 \times 7^2 \times 11^1 \times 13^1 \times 17^1 \times 19^1 | 4608 |
| 20 | 195643523275200 | $2^7 \times 3^5 \times 5^2 \times 7^2 \times 11^1 \times 13^1 \times 17^1 \times 19^1 | 6912 |