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Hyperfactorial

The hyperfactorial (Sloane and Plouffe 1995) is the function defined by(1)(2)where is the K-function.The hyperfactorial is implemented in the WolframLanguage as Hyperfactorial[n].For integer values , 2, ... are 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (OEIS A002109).The hyperfactorial can also be generalized to complex numbers, as illustrated above.The Barnes G-function and hyperfactorial satisfy the relation(3)for all complex .The hyperfactorial is given by the integral(4)and the closed-form expression(5)for , where is the Riemann zeta function, its derivative, is the Hurwitz zeta function, and(6) also has a Stirling-like series(7)(OEIS A143475 and A143476). has the special value(8)(9)(10)where is the Euler-Mascheroni constant and is the Glaisher-Kinkelin constant.The derivative is given by(11)..

Ultraprimorial

The function defined by , where is a prime number and is a primorial. The values for , 3, ..., are 4, 46656, 205891132094649000000000000000000000000000000, ... (OEIS A165812).

Triangle coefficient

The triangle coefficient is function of three variables written and defined by(Shore and Menzel 1968, p. 273), where is a factorial. It arises for example in the definition of the Wigner 6j-symbol.

Multifactorial

A generalization of the factorial and doublefactorial,(1)(2)(3)etc., where the products run through positive integers.The following table lists the values of the first few multifactorials for , 2, ....A0001421, 2, 6, 24, 120, 720, ...A0068821, 2, 3, 8, 15, 48, 105, ...A0076611, 2, 3, 4, 10, 18, 28, 80, 162, 280, ...A0076621, 2, 3, 4, 5, 12, 21, 32, 45, 120, ...

Superfactorial

The superfactorial of is defined by Pickover (1995) as(1)The first two values are 1 and 4, but subsequently grow so rapidly that already has a huge number of digits.Sloane and Plouffe (1995) define the superfactorial by(2)(3)which is equivalent to the integral values of the Barnes G-function. The values for , 2, ... are 1, 1, 2, 12, 288, 34560, ... (OEIS A000178). This function has an unexpected connection with Bell numbers.

Double factorial

The double factorial of a positive integer is a generalization of the usual factorial defined by(1)Note that , by definition (Arfken 1985, p. 547).The origin of the notation appears not to not be widely known and is not mentioned in Cajori (1993).For , 1, 2, ..., the first few values are 1, 1, 2, 3, 8, 15, 48, 105, 384, ... (OEIS A006882). The numbers of decimal digits in for , 1, ... are 1, 4, 80, 1285, 17831, 228289, 2782857, 32828532, ... (OEIS A114488).The double factorial is implemented in the WolframLanguage as n!! or Factorial2[n].The double factorial is a special case of the multifactorial.The double factorial can be expressed in terms of the gammafunction by(2)(Arfken 1985, p. 548).The double factorial can also be extended to negative odd integers using the definition(3)(4)for , 1, ... (Arfken 1985, p. 547). Min Max Re Im Similarly, the double factorial can be extended to complex arguments as(5)There are many identities..

Falling factorial

The falling factorial , sometimes also denoted (Graham et al. 1994, p. 48), is defined by(1)for . Is also known as the binomial polynomial, lower factorial, falling factorial power (Graham et al. 1994, p. 48), or factorial power.The falling factorial is related to the rising factorial (a.k.a. Pochhammer symbol) by(2)The falling factorial is implemented in the Wolfram Language as FactorialPower[x, n].A generalized version of the falling factorial can defined by(3)and is implemented in the Wolfram Language as FactorialPower[x, n, h].The usual factorial is related to the falling factorialby(4)(Graham et al. 1994, p. 48).In combinatorial usage, the falling factorial is commonly denoted and the rising factorial is denoted (Comtet 1974, p. 6; Roman 1984, p. 5; Hardy 1999, p. 101), whereas in the calculus of finite differences and the theory of special functions, the falling factorial is denoted..

Factorion

A factorion is an integer which is equal to the sum of factorials of its digits. There are exactly four such numbers:(1)(2)(3)(4)(OEIS A014080; Gardner 1978, Madachy 1979, Pickover 1995). Obviously, the factorion of an -digit number cannot exceed .

Rising factorial

The rising factorial , sometimes also denoted (Comtet 1974, p. 6) or (Graham et al. 1994, p. 48), is defined by(1)This function is also known as the rising factorial power (Graham et al. 1994, p. 48) and frequently called the Pochhammer symbol in the theory of special functions. The rising factorial is implemented in the Wolfram Language as Pochhammer[x, n].The rising factorial is related to the gamma function by(2)where(3)and is related to the falling factorial by(4)The usual factorial is therefore related to the risingfactorial by(5)for nonnegative integers (Graham et al. 1994, p. 48).Note that in combinatorial usage, the falling factorial is denoted and the rising factorial is denoted (Comtet 1974, p. 6; Roman 1984, p. 5; Hardy 1999, p. 101), whereas in the calculus of finite differences and the theory of special functions, the falling factorial is denoted and the rising factorial is denoted..

Pochhammer symbol

The Pochhammer symbol(1)(2)(Abramowitz and Stegun 1972, p. 256; Spanier 1987; Koepf 1998, p. 5) for is an unfortunate notation used in the theory of special functions for the rising factorial, also known as the rising factorial power (Graham et al. 1994, p. 48) or ascending Factorial (Boros and Moll 2004, p. 16). The Pochhammer symbol is implemented in the Wolfram Language as Pochhammer[x, n].In combinatorics, the notation (Roman 1984, p. 5), (Comtet 1974, p. 6), or (Graham et al. 1994, p. 48) is used for the rising factorial, while or denotes the falling factorial (Graham et al. 1994, p. 48). Extreme caution is therefore needed in interpreting the notations and .The first few values of for nonnegative integers are(3)(4)(5)(6)(7)(OEIS A054654).In closed form, can be written(8)where is a Stirling number of the first kind.The Pochhammer symbol satisfies(9)the dimidiation formulas(10)(11)and..

Factorial sums

The sum-of-factorial powers function is defined by(1)For ,(2)(3)(4)where is the exponential integral, (OEIS A091725), is the En-function, is the real part of , and i is the imaginary number. The first few values are 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, ... (OEIS A007489). cannot be written as a hypergeometric term plus a constant (Petkovšek et al. 1996). The only prime of this form is , since(5)(6)(7)is always a multiple of 3 for .In fact, is divisible by 3 for and , 5, 7, ... (since the Cunningham number given by the sum of the first two terms is always divisible by 3--as are all factorial powers in subsequent terms ) and so contains no primes, meaning sequences with even are the only prime contenders.The sum(8)does not appear to have a simple closed form, but its values for , 2, ... are 1, 5, 41, 617, 15017, 533417, 25935017, ... (OEIS A104344). It is prime for indices 2, 3, 4, 5, 7, 8, 10, 18, 21, 42, 51, 91, 133, 177, 182, 310, 3175, 9566, 32841,..

Alternating factorial

The alternating factorial is defined as the sum of consecutive factorialswith alternating signs,(1)They can be given in closed form as(2)where is the exponential integral, is the En-function, and is the gamma function.The alternating factorial will is implemented in the WolframLanguage as AlternatingFactorial[n].A simple recurrence equation for is given by(3)where .For , 2, ..., the first few values are 1, 1, 5, 19, 101, 619, 4421, 35899, ... (OEIS A005165).The first few values for which are (probable) primes are , 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164, 43592, 59961, ... (OEIS A001272; extending Guy 1994, p. 100). Živković (1999) has shown that the number of such primes is finite. was verified to be prime in Jul. 2000 by team of G. La Barbera and others using the Certifix program developed by Marcel Martin.The following table summarizes the largest..

Factorial prime

A factorial prime is a prime number of the form , where is a factorial. is prime for , 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 26951, 34790, 94550, 103040, 147855, 208003, ... (OEIS A002982), the largest of which are summarized in the following table.digitsdiscoverer107,707Marchal, Carmody, and Kuosa (Caldwell; May 2002)142,891Marchal, Carmody, and Kuosa (Caldwell; May 2002)429,390D. Domanov/PrimeGrid (Oct. 4, 2010)471,794J. Winskill/PrimeGrid (Dec. 14, 2010)700,177PrimeGrid (Aug. 30, 2013)1,015,843S. Fukui (Jul. 25, 2016; https://primes.utm.edu/primes/page.php?id=121944) is prime for , 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, ... (OEIS A002981; Wells 1986, p. 70), the largest of which are summarized in the following table.digitsdiscoverer107,707K. Davis..

Double factorial prime

A double factorial prime is a prime number of the form , where is a double factorial. is prime for , 4, 6, 8, 16, 26, 64, 82, 90, 118, 194, 214, 728, ... (OEIS A007749), the largest of which are summarized in the following table.digitsdiscoverer169,435S. Fukai (Jun. 5, 2015)229,924S. Fukai (Jun. 5, 2015)344,538S. Fukai (Apr. 21, 2016) is prime for , 1, 2, 518, 33416, 37310, 52608, 123998, ... (OEIS A080778), the largest of which are summarized in the following table.digitsdiscoverer112,762H. Jamke (Jan. 3, 2008)288,864S. Fukai (Jun. 5, 2015)

Factorial

The factorial is defined for a positive integer as(1)So, for example, . An older notation for the factorial was written (Mellin 1909; Lewin 1958, p. 19; Dudeney 1970; Gardner 1978; Conway and Guy 1996).The special case is defined to have value , consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set ).The factorial is implemented in the Wolfram Language as Factorial[n] or n!.The triangular number can be regarded as the additive analog of the factorial . Another relationship between factorials and triangular numbers is given by the identity(2)(K. MacMillan, pers. comm., Jan. 21, 2008).The factorial gives the number of ways in which objects can be permuted. For example, , since the six possible permutations of are , , , , , . The first few factorials for , 1, 2, ... are 1, 1, 2, 6, 24, 120, ... (OEIS A000142).The..

Subfactorial

The th subfactorial (also called the derangement number; Goulden and Jackson 1983, p. 48; Graham et al. 2003, p. 1050) is the number of permutations of objects in which no object appears in its natural place (i.e., "derangements").The term "subfactorial "was introduced by Whitworth (1867 or 1878; Cajori 1993, p. 77). Euler (1809) calculated the first ten terms.The first few values of for , 2, ... are 0, 1, 2, 9, 44, 265, 1854, 14833, ... (OEIS A000166). For example, the only derangements of are and , so . Similarly, the derangements of are , , , , , , , , and , so .Sums and formulas for include(1)(2)(3)(4)where is a factorial, is a binomial coefficient, and is the incomplete gamma function.Subfactorials are implemented in the WolframLanguage as Subfactorial[n].A plot the real and imaginary parts of the subfactorial generalized to any real argument is illustrated above, with the usual integer-valued subfactorial..

Roman factorial

(1)The Roman factorial arises in the definition of the harmonic logarithm and Roman coefficient. It obeys the identities(2)(3)(4)where(5)and(6)

Primorial prime

Primorial primes are primes of the form , where is the primorial of . A coordinated search for such primes is being conducted on PrimeGrid. is prime for , 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, ... (OEIS A057704; Guy 1994, pp. 7-8; Caldwell 1995). These correspond to with , 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, ... (OEIS A006794). The largest known primorial primes as of Nov. 2015 are summarized in the following table (Caldwell).digitsdiscoverer6845Dec. 1992365851PrimeGrid (Dec. 20, 2010)476311PrimeGrid (Mar. 5, 2012) (also known as a Euclid number) is prime for , 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, ... (OEIS A014545; Guy 1994, Caldwell 1995, Mudge 1997). These correspond to with , 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547,..

Primorial

Let be the th prime, then the primorial (which is the analog of the usual factorial for prime numbers) is defined by(1)The values of for , 2, ..., are 2, 6, 30, 210, 2310, 30030, 510510, ... (OEIS A002110).It is sometimes convenient to define the primorial for values other than just the primes, in which case it is taken to be given by the product of all primes less than or equal to , i.e.,(2)where is the prime counting function. For , 2, ..., the first few values of are 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, ... (OEIS A034386).The logarithm of is closely related to the Chebyshev function , and a trivial rearrangement of the limit(3)gives(4)(Ruiz 1997; Finch 2003, p. 14; Pruitt), where eis the usual base of the natural logarithm.

Factorial products

The first few values of (known as a superfactorial) for , 2, ... are given by 1, 2, 12, 288, 34560, 24883200, ... (OEIS A000178).The first few positive integers that can be written as a product of factorials are1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, ... (OEIS A001013).The number of ways that is a product of smaller factorials, each greater than 1, for , 2, ... is given by 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, ... (OEIS A034876), and the numbers of products of factorials not exceeding are 1, 2, 4, 8, 15, 28, 49, 83, ... (OEIS A101976).The only known factorials which are products of factorials in an arithmeticprogression of three or more terms are(1)(2)(3)(Madachy 1979).The only solutions to(4)are(5)(6)(7)(Cucurezeanu and Enkers 1987).There are no nontrivial identities of the form(8)for with for for except(9)(10)(11)(12)(Madachy 1979; Guy 1994, p. 80). Here, "nontrivial" means that identities with , or equivalently are excluded, since..

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