Log hyperfactorial. Remember to check what links here and the page history before deleting. 埃尔米特 (Hermite) 多项式的判别式和超阶乘相关： Support this channel and get more content 👉 / blackpenredpen Hyperfactorial, • Hyperfactorial introduction Note: on Wikipedia, it uses sf (n) for the Sloane and plouffe’s definition of the Oct 6, 2014 · The hyperfactorial function is similar to the factorial, but produces larger numbers. Feb 16, 2021 · Hyperfactorial of a number is obtained by multiplying consecutive integers from 1 to the given number, each raised to its on power. May 9, 2023 · T (n) = T (n-i) + log (n-i) + log (n-i-1) + log (n), and I can say n-i = 1 But I am having trouble figuring out what to do when T (n-2), and how that affects nlogn. . The FM broadcast band ends at 108 MHz in most countries (except Japan, where the frequency range 99-108 MHz is reserved for digital broadcasting), but in Abstract: The special factorial functions is an extension of the general factorial notation which is the concept of this paper. This paper provides the relationship between the subfactorial, superfactorial, hyperfactorial, G – function, K – function, and the gamma function. What is a hyperfactorial? If you like factorials, then check out "7 factorials you probably didn't know" • 7 factorials you probably didn't know 🛍 Shop my math t-shirt & hoodies: amzn. 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. Abstract In this article, we investigate the p 𝑝 p italic_p -adic valuation ν p subscript 𝜈 𝑝 \nu_ {p} italic_ν start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT of quantities such as the factorial n! 𝑛 n! italic_n !, the hyperfactorial H (n) 𝐻 𝑛 H (n) italic_H ( italic_n ) or the superfactorial sf (n) sf 𝑛 \mathrm {sf} (n) roman_sf ( italic_n ). Exponential Factorial Annoyingly also denoted the exponential factorial is like a normal factorial but exponentiated instead of multiplied: Hyperfactorial Finally the hyperfactorial is defined like this: Or: And hence is very similar to the Sloane definition of the superfactorial: I've been familiarizing myself with the hyperfactorial, and I'm simply curious if it has an extension/analogue into the world of rational numbers, irrational numbers, and complex numbers like the c The reason given is: This has been moved to "hyperfactorial array notation numbers" on the "list of nonnotable numbers" page. Hyperfactorial automatically threads over lists. " Jun 23, 2025 · Given a number, the task is to find the hyperfactorial of a number. 2 108 is the largest known power of two not containing digit 9. Logarithm rules and propertiesLogarithm Rules The base b logarithm of a number is the exponent that we need to raise the base in order to get the number. If you disagree with its deletion, please explain why at Category talk:Candidates for deletion or improve the page and remove the {{delete}} tag. 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. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. 666108 is the first power of 666 larger than a centillion. Fur Affinity | For all things fluff, scaled, and feathered! Hyperfactorial [n] 给出超阶乘函数 Hyperfactorial [n]. for hyperoperators x {n}y also works for hyperoperators x! for factorials FGH: f_n (x). "Stirling's Series. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. That is,H (n) = 1^1\cdot 2^2\cdot \cdots n^n = \prod_ {i=1}^ {n} i^i = n^n H (n-1). AI explanations are generated using OpenAI technology. Apr 18, 2018 · A brief introduction to two interesting variations of the factorial function: the superfactorial and hyperfactorial. to hyperfactorial (plural hyperfactorials) (mathematics) The result of multiplying a given number of consecutive integers from 1 to the given number, each raised to its own power. Arfken, G. Logarithm definition Logarithm rules Logarithm problems Complex logarithm Graph of log (x) Logarithm table Logarithm calculator Logarithm definition When b is raised to the power of y is equal x: b y = x Then the base b logarithm of x is List of functions Operations: x+y, x-y, x*y, x/y, x^y x^^y, x^^^y, etc. n can be any number, 'w', or 'w+1'. 2108 is the largest known power of two not containing digit 9. 6 days ago · 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. 666 108 is the first power of 666 larger than a centillion. In mathematics, and more specifically number theory, the hyperfactorial of a positive integer n is the product of the numbers of the form x^x from 1^1 to n^n. 257, 1972. The hyperfactorial of a positive integer is the product of the numbers . [edit 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: Specific numbers 108 is the third hyperfactorial number. This video demonstrates how to compute the hyperfactorial for This free log calculator solves for the unknown portions of a logarithmic expression using base e, 2, 10, or any other desired base. A. Following the usual convention for the empty 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. (Eds. In combination of these functions, formulas were derived for finding each of the special factorial functions listed Nested Factorials: Tetorial · Petorial · Ectorial · Zettorial · Yottorial Array-based extensions: Hyperfactorial array notation · Nested factorial notation Other googological variants: · Tetrofactorial · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial Categories. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music… 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: Solve problems, write better, and save 42% on your first year. The FM broadcast band ends at 108 MHz in most countries (except Japan, where the frequency range 99-108 MHz is reserved for digital broadcasting), but in most 6 days ago · Bernoulli Number, Gamma Function, K-Function, Log Gamma Function, Permutation Cycle, Stirling's Approximation Explore with Wolfram|Alpha References Abramowitz, M. […] Details Mathematical function, suitable for both symbolic and numeric manipulation. The rate of growth of this function, however, is not much larger than a regular factorial. Definition The hyperfactorial of a positive integer n is the product of the numbers 1^1, 2^2, \dots, n^n. Functions: log10 (x) for log base 10 ln (x) for natural log ackermann (x) for the 1-input Ackermann function approximateFGH (x) to get the approximate FGH equivalent of a number It grows insanely fast. Jul 11, 2025 · Given a number, the task is to find the hyperfactorial of a number. New York: Dover, p. Hyperfactorial can be evaluated to arbitrary numerical precision. This result plays a key role in the analysis of the computational complexity of sorting algorithms (see comparison sort). 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). and Stegun, I. This function is a shifted version of the superfactorial (Sloane and In mathematics, and more specifically number theory, the hyperfactorial of a positive integer n is the product of the numbers of the form x x from 1 1 to n n. AI generated content may present inaccurate or offensive content that does not represent Symbolab's view. Simplify logarithmic expressions using algebraic rules step-by-step. That is, [1][2] Following the usual convention for the empty product, the hyperfactorial of 0 is 1. H(n)= 1 ^ 1 * 2 ^ 2 * 3 ^ 3 * . The hyperfactorial function satisfies . It is also considered sacred by the Dharmic religions. ). Apr 4, 2009 · A simple approximation for log n! based on Stirling's approximation is A much better approximation for log n! was given by Srinivasa Ramanujan [citation needed]: One can see from this that log n! is Ο (n log n). In particular, we obtain Specific numbers [edit] 108 is the third hyperfactorial number. Hyperfactorial is defined as for positive integers and is otherwise defined as . The hyperfactorial is implemented in the Wolfram Language as Hyperfactorial [n]. w4wfi m4lmh2 zcx4 vts6r u7rva gmihh 1rk r02rzn2 v66dkmmp nbj2dn