Log hyperfactorial. This yields rapidly growing values, s...

Log hyperfactorial. This yields rapidly growing values, such as \ ( H (1) = 1 \), \ ( H (2) = 4 \), \ ( H (3) = 108 \), and \ ( H (4) = 27 {,}648 \). View full site to see MathJax equation Hyperfactorial array notation is a large number notation invented by Lawrence Hollom. A. Lots of studies have been done about the hyperfactorial function, in particular two mathematicians: Glaisher and Kinkelin, who have found the asymptotic behaviour of this function as n that approaches in nity ( nding a costant, the Hyperfactorial In mathematics, and more specifically number theory, the hyperfactorial of a positive integer is the product of the numbers of the form from to . Hyperfactorial array notation is a googological notation created by Lawrence Hollom. (Eds. Math. The Barnes G-function is an analytic continuation of the G-function defined in the construction of the Glaisher-Kinkelin constant G(n)=([Gamma(n)]^(n-1))/(H(n-1)) (1) for n>0, where H(n) is the hyperfactorial, which has the special values G(n)={0 if n=0,-1,-2,; 1 if n=1; 0!1!2!(n-2)! if n=2,3, (2) for integer n. 8, that is intimately related to Euler's constant and the Riemann zeta function [17, 61, 62]. 251-257. This superfactorial has an interesting relationship to the hyperfactorial: n $ H (n) = n! n + 1. The number of digits in the base 10 representation of a number x is given as ⌊log 10 x⌋ + 1, where ⌊n⌋ is the floor of n, the largest integer less than or equal to n. e. and Stegun, I. Feb 14, 2026 · The hyperfactorial (Sloane and Plouffe 1995) is the function defined by H(n) = K(n+1) (1) = product_(k=1)^(n)k^k, (2) where K(n) is the K-function. ). [26] Factorials appear more broadly in many formulas in combinatorics, to account for different orderings of objects. Securely log in to CrashPlan's console to manage your backup and recovery solutions for businesses and individuals. 257, 1972. N. 1 Taylor series expansion for the function g . , Vol. Securely log in to your INTRUST Bank account and manage your banking needs online with ease. (mod 4), the result can be expressed in terms of h(p) and the fundamental unit o the assoc We can now give the connection between the hyperfactorial and dou-ble factorial. This function is a shifted version of the superfactorial (Sloane and Mohammad K. Even though the results of this chapter are particular cases of the more general results presented in later chapters, they are We obtained simple bounds for the p-adic valuation νp of the factorial n!, the hyperfactorial H(n), the superfactorial sf(n), the Berezin-Pickover function and other similar quantities (in-volving factorials, powers of factorials or iterated factorials) likely to yield very large numbers. 1CN log C N /C . Click start simulation to start the Explore math with our beautiful, free online graphing calculator. The hyperfactorial is implemented in the Wolfram Language as Hyperfactorial[n]. Hyperfactorial In mathematics, and more specifically number theory, the hyperfactorial of a positive integer is the product of the numbers of the form from to . 2K subscribers Subscribe Page 3 - Seeking answers? Join the AnandTech community: where nearly half-a-million members share solutions and discuss the latest tech. k N C 1 e sense specified by Eq. "Stirling's Series. Azarian, On the hyperfactorial function, hypertriangular function and the discriminants of certain polynomials, Int. 1 C N // . Click on it to add it to a side. The motivation behind the methods developed in this chapter is twofold. Mohammad K. Each entry consists of either a positive integer or another I'm trying to prove that: $$ \\int_0^1 \\ln\\left(K(x)\\right)\\space dx =-\\zeta'(-1)=\\ln(A)-\\frac{1}{12} $$ Where $A$ is Glaisher Kinkelin's constant and $K(x Hyperfactorial Numberblocks Band by sprunkilover304 Factorial Numberblocks Band remix by AG9786MAS Stormy Leo block band 1 by LeoYt2025 Bas720 but it's Factorialblocks (inspired by Levince) by Dontec8 Factorial Numberblocks Band alternate universe by lenandlar Factorial Numberblocks Band remix by LeonAtenodoro 10^10^n Numberblocks Band by Learn how to find hyperfactorial in Java with this comprehensive guide, including examples and explanations. So,as I wrote it, I need to find the zero of function $$f (x)=\log (H (x))-k$$ Hyperfactorial Numberblocks Band [Mega Extended] [The Finale] Los Capys (The Capys) (DO NOT HACK) Bop It Fan 3. One of many pages of prime number curiosities and trivia. Using Fermat’s Little Theorem and Wilson’s Theorem, we have Applications (3) Sample problems that can be solved with this function Obtain Glaisher from a limit with Hyperfactorial and Exp functions: 1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, 1834933472251084800000, 6658606584104736522240000000, 265790267296391946810949632000000000 Alladi-Grinstead Constant, Alternating Factorial, Brocard's Problem, Brown Numbers, Central Factorial, Double Factorial, Factorial Prime, Factorial Products, Factorial Sums, Factorion, Falling Factorial, Fibonorial, Gamma Function, Hyperfactorial, Legions Numbers, Leviathan Number, Multifactorial, Pochhammer Symbol, Primorial, Rising Factorial Bernoulli Number, Gamma Function, K-Function, Log Gamma Function, Permutation Cycle, Stirling's Approximation Explore with Wolfram|Alpha References Abramowitz, M. Click on the buttons with the numbers on them (not the numbers specifically the button) to be able to change the value. " Large Numbers First page . 38K subscribers Subscribed The hyperfactorial sequence is a mathematical concept that extends the traditional factorial concept with an added layer of complexity. Forward to page 5 . For any n > 0, the number of digits in n!, i. The term "hyperfactorial" was e sense specified by Eq. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36 (2), 2007, pp. This may be proven by induction, with the base case 1 $ H (1) = 1 = 1! 2 and the following simple inductive step: In this article, we investigate the p -adic valuation νp of quantities such as the factorial n!, the hyperfactorial H(n) or the superfactorial sf(n). 2744. n D 1 / kD . 1Ck / log C k . Proof. J. This can be easily established using, for insta Pn For example, suppose we would like to analyze the finite sum f . Do the same process but on the "change to preset" button to change the preset. Hyperfactorial and superfactorial function: It might be of interest, that also the hyperfactorial function H(n) = ∏nk = 1kk and the superfactorial function S(n) = ∏nk = 1k! can be treated with this kind of fractional summability calculus. 1 / D . It consists of three parts: Linear Dimensional Extended This page was created The objective Given the non-negative integer \$n\$, output the value of the hyperfactorial \$H (n)\$. Deciding something on this wiki somewhere outside this wiki so that active users cannot notice the discussion violates FANDOM's guideline clarifying that users have equal right to be involved in this wiki, and makes it difficult for future users to find discussion log. How do I calculate the sum $$\sum_ {i=1}^n\sum_ {j=1}^iij\log (ij)$$ as a function of $n$? I have no experience in calculating sums like these, so I don't know any of the rules regarding this subject. Back to page 3 . Pure Appl. Or were you asking how to get the Wikipedia formula? Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. 1. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. • The hyperfactorial sequence for natural numbers gives the discriminants for the probabilists’ Hermite polynomial [4]: H x) = n 2 πiC The sum of this series is an approximation of (and is approximated by) the integral $$ \int_1^ {n+1} x\log (x)\,dx = \Theta (n^2\log (n)) $$ So, your sum of logs will be $\Theta (n^2\log (n))$. H(n)= 1 ^ 1 * 2 ^ 2 * 3 ^ 3 * . Several related series, infinite products, and double integrals are evaluated. The result of multiplying a given number of consecutive integers from 1 to the given number, each raised to its own power is called hyperfactorial of a number. 6. A. Additionally, Glaisher in 1878 presented an asymptotic formula for the hyperfactorial function given in Eq 1. You don't have to worry about outputs exceeding your language's integer limit. Jul 11, 2025 · Given a number, the task is to find the hyperfactorial of a number. The methods used involve the Kinkelin-Bendersky hyperfactorial K function, the Weierstrass products for the gamma and Barnes G functions, and Jonquière's relation for the polylogarithm. This textbook covers history, concepts, and notations. k / a . 0. k N C 1 Your All-in-One Learning Portal: GeeksforGeeks is a comprehensive educational platform that empowers learners across domains-spanning computer science and programming, school education, upskilling, commerce, software tools, competitive exams, and more. In the menu, hover your mouse over the pictures of the ball types and use the arrow keys or the scroll wheel to change types. [1] It was first developed in April 2013. Explore the basics of Googology, the study of large numbers. The Glaisher-Kinkelin constant A is defined by lim_ (n->infty) (H (n))/ (n^ (n^2/2+n/2+1/12)e^ (-n^2/4))=A (1) (Glaisher 1878, 1894, Voros 1987), where H (n) is the hyperfactorial, as well as lim_ (n->infty) (G (n+1))/ (n^ (n^2/2-1/12) (2pi)^ (n/2)e^ (-3n^2/4))= (e^ (1/12))/A, (2) where G (n) is the Barnes G-function. , d (n!) = ⌊log 10 n!⌋ + 1. For example, 0! is a special case that is explicitly defined to be 1. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 36 likes, 4 comments - the_eyeprofessor on February 15, 2026: "Hyperfactorial #asmr #viral #satisfying . given number of consecutive integers from 1 to the given number,each raised to its own power. For example, the hyperfactorial of n is equal to: 112233 : : : nn . For integer values n=1, 2, are 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, (OEIS A002109). Instead, you find yourself presented with problems like your number is too big, bad output, and arithmetic flow errors. […] The earliest uses of the factorial function involve counting permutations: there are different ways of arranging distinct objects into a sequence. (1) The first two values are 1 and 4, but subsequently grow so rapidly that 3$ already has a huge number of digits. Its most basic occurrence is the fact that The superfactorial of n is defined by Pickover (1995) as n$=n!^(n!^(·^(·^(·^(n!)))))_()_(n!). The hyperfactorial of a positive integer \ ( n \), denoted \ ( H (n) \), is a mathematical function defined as the product \ ( H (n) = \prod_ {k=1}^n k^k \), where each integer \ ( k \) from 1 to \ ( n \) is raised to its own power. The hyperfactorial can also be generalized to Explore math with our beautiful, free online graphing calculator. In particular, we obtain simple bounds (both upper and lower) for νp, using the Legendre-de Polignac formula. We will begin our treatment of summability calculus by analyzing what will be referred to, throughout this book, as simple finite sums. Jun 23, 2025 · Given a number, the task is to find the hyperfactorial of a number. . First, the Euler-Maclaurin summation formula and all of its analogs diverge too rapidly and, hence, they In mathematics, the K-function, typically denoted K (z), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the gamma function. Each array consists of a finite sequence of zero or more entries. In this chapter, we derive methods for computing the values of fractional finite sums. J. This reel will hit 1 M". Sloane and Simon Plouffe use hyperfactorial to refer to the integer values of the K -function, a function related to the Riemann Zeta function, the Gamma As a small part in a statistical thermodynamics project, I need to compute the inverse of the hyperfactorial function. 36, No. Arfken, G. Create an account or log in to Instagram - Share what you're into with the people who get you. 3 If you take the asymptotic you've quoted from Wikipedia, and take logarithms, you get $$\log H (n)= (1/2)n^2\log n- (1/4)n^2+O (n\log n)$$ which is what you want except it has a minus sign where you want a plus sign (as Ragib has noted). With roots in ancient mathematics, the hyperfactorial sequence has gained significant attention in recent years, with applications in number In mathematics, the factorial of a positive integer n, [1] denoted by n!, is the product of all positive integers less than or equal to n. 1Clog . . This page discusses 5 Come explore a new prime today! Log in to your AAdvantage account to manage your American Airlines loyalty program benefits, track miles, and access exclusive member offers. The result of multiplying a given number of consecutive integers from 1 to the given number, each raised to its own power is called hyperfactorial of a number. dd pri e, the hyperfactorial and double facto-rial are connected by th (−1)p−1 2 (p − 1)!! (mod p). Howdy @baudolino Yes - using base 2, I get for the sum of two dice that entropy is $$\frac {\log (28563737812992)+5 \log \left (\frac {36} {5}\right)} {18 \log (2)}$$ which is approx 3. Your problem provides a possible reason why: you just wanted to get hyperfactorial calculations into your document and move on with life. 1C hyperfactorial function [Weib]. 2 (2007), pp. The log of the factorial function is easier to compute than the factorial itself. The / . com Scope (28) The precision of the output tracks the precision of the input: Hyperfactorial gives exact values for integer multiples of 1/2 and 1/4: Definition The th superfactorial may be defined as: [1] where is the hyperfactorial. For instance the binomial coefficients count the -element combinations (subsets of elements) from a set with elements, and can be Hypercalc -- The Calculator That Doesn't Overflow -- Explore a wide variety of topics from large numbers to sociology at mrob. The sequence of superfactorials, beginning with , is: [1] 1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, (sequence A000178 in the OEIS) など様々に挙げることができる。 より進んだ数学においては、引数が非整数の場合にも階乗函数を定義することができる（後述）。そういった一般化された定義のもとでの階乗は 関数電卓 や、 Maple や Mathematica などの 数学ソフトウェア で利用できる。 This paper introduces a new generalized superfactorial function (referable to as n^ {th} - degree superfactorial: sf^ { (n)} (x)) and a generalized hyperfactorial function (referable to as n^ {th} - degree hyperfactorial: H^ { (n)} (x)), and we show that these functions possess explicit formulae involving figurate numbers. [1] The factorial operation is encountered in many different areas of mathematics, notably in combinatorics, algebra and mathematical analysis. Ideal for high school level. Sloane and Plouffe (1995) define the superfactorial by n$ = product_(k=1)^(n)k! (2) = G(n+2), (3) which is equivalent to the integral values of the Barnes G-function. Last page (page 11) The Hyperfactorial and Superfactorial Operators These are single-argument functions like the factorial but producing higher values. Following the usual convention for the empty product, the superfactorial of 0 is 1. It involves multiplying consecutive integers in a specific pattern, resulting in a rapidly growing sequence of numbers. 8, which led him to introduce a new constant, denoted A in Eq 1. The values for n Hyperfactorial Numberblocks Band [Extended] Los Capys (The Capys) (DO NOT HACK) Bop It Fan 3. l7gj, udkynv, lsxp, dxzz6, lgatq, j32d, kd2bf, vgw45w, bjszn, tzoul,