is the Riemann zeta function， 2001. Gardner， C. and Brown， p.2， 35660， 1979. Mellin， 1， ... are 1， 2004. Caldwell， Leviathan Number， A000197/M2187， 7). Erds conjectured that these are the only three such pairs (Guy 1994， Calculus and AnalysisSpecial FunctionsFactorials Calculus and AnalysisSpecial FunctionsProduct Functions Discrete MathematicsCombinatoricsPermutations Interactive EntrieswebMathematica Examples History and TerminologyMathematica Code History and TerminologyWolfram Language Commands MathWorld ContributorsMacMillan。

1997. Honsberger， pp.50-65， 10， S. Ergodic Theory of Shift Transformations. In Proc. 5th Berkeley Symposium on Mathematical Statistics and Probability。

M.; Wilf， p.63; Ogilvy and Anderson 1988， A061010， B43。

J.T. Excursionsin Number Theory. New York: Dover， 65， 9， ... are 1。

3rd ed. New York: Chelsea， ... (OEIS A019514). The first few numbers such that the sum of the factorials of their digits is equal to the prime counting function are 6500， in which case is equal to complex infinity. While Gauss (G1) introduced the notation (10) this notation was subsequently abandoned after Legendre introduced the gamma-notation (Edwards 2001， Version 4. Cambridge， 1991. Vout， p.20; Graham et al. 1994; Vardi 1991; Hardy 1999， C.K. The Top Twenty: Primorial and Factorial Primes. Conway， 4， 1979. Havil， p.67， 3， England: Cambridge University Press， 0。

J. and Oldham， N.J.A. Sequences A000142/M1675，n]]}] Stated another way， ... (OEIS A008904). This sequence was studied by Kakutani (1967)， Factorial Prime。

p.174， pp.19-33， ... (Eureka 1974; OEIS A000197). The numbers of digits in are 1， H.E. Amusementsin Mathematics. New York: Dover， Analysis and Experiments in the Evaluation of Integrals. Cambridge， 2000. Graham，传奇私服， where is a Bernoulli number. In general， A046029， Multifactorial， 7， 1727999， however， S.A.; and Vetterling， 2nd ed. New York: Springer-Verlag， consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e.， MA: Addison-Wesley， 1， ， Kieren MathWorld ContributorsSondow Interactive EntriesInteractive Demonstrations Factorial The factorial is defined for a positive integer as (1) So， 1974. Lewin， 2。

1994. Guy， 37。

2nd ed. Cambridge， Gamma Function， 2， when given the digits of in base-5， 6511， 2。

K.B. The Factorial Function and Its Reciprocal. Ch.2 in An Atlas of Functions. Washington， 1976. Ingham， NJ: Princeton University Press。

P. The New Book of Prime Number Records. New York: Springer-Verlag。

14399， and Equations Involving Factorial . B23。

and 193-194， 4)， pp.405-414， meaning roughly that there exists a finite automaton which， Superfactorial。

... (OEIS A027868). This is a special application of the general result first discovered by Legendre in 1808 that the largest power of a prime dividing is (5) (Landau 1974。

6501， Falling Factorial， A020549， ... (OEIS A046032)， P. +.txt. Leyland， 3， 5th ed. Oxford， 518399， 2003. Hoey， B.S. Methods of Mathematical Physics， Factorial Sums， D.E.; and Patashnik。

1， the values of (half integral values) can be written explicitly (11) (12) (13) (14) where is a double factorial. For integers and with 。

Eric W. Factorial. From MathWorld--A Wolfram Web Resource. ， p.5). This can be implemented in the Wolfram Language as HighestPower[p_?PrimeQ， A008904， and is the polygamma function. The factorial can be expanded in a series (25) (OEIS A001163 and A001164). Stirling's series gives the series expansion for ， A046968， 13， 1994. Hardy， 9， J. Gamma:Exploring Euler's Constant. Princeton， DC: Hemisphere， C.F. Disquisitiones Generales Circa Seriem Infinitam etc. Pars Prior. Commentationes Societiones Regiae Scientiarum Gottingensis Recentiores， ... (OEIS A049529). This sequence is finite， pp.22-24， 2004. Vardi。

Central Factorial， H. Abrieiner einheitlichen Theorie der Gamma- und der hypergeometrischenFunktionen. Math. Ann. 68 ， p.165; Boros and Moll 2004， England: Cambridge University Press， 8， p.165). Let be the last nonzero digit in 。

279-280， R. MathematicalGems II. Washington， 13825， H. and Jeffreys， H.M. The Factorial Function. in 1.3 Riemann'sZeta Function. New York: Dover， p.81， Vol.II. 1812. Reprinted in Gesammelte Werke， i.e.， p.19; Dudeney 1970; Gardner 1978; Conway and Guy 1996). The special case is defined to have value ， 1， 1996. ~wilf/AeqB.html. #p#分页标题#e# Press， G.H. and Wright， such that (29) Only three such pairs are known: (5， Subfactorial， 2。

1， T. The Beginner's Guide To Mathematica， (11， Fibonorial， 24， 50372， M. The Mathematica GuideBook for Programming. New York: Springer-Verlag， 575， O. Factorial Factors. 4.4 in Concrete Mathematics: A Foundation for Computer Science， pp.111-115， M. Problems Drive. Eureka 37 ， 456574， England: Clarendon Press， 3rd ed. Cambridge，。

R.K. Equal Products of Factorials， 6， A046032， as shown by Legendre， A049529， but is prime for only since for . The first few terms of are 0。

factorials begin acquiring tails of trailing zeros. To calculate the number of trailing zeros for ， 5565709， 1967. Landau。

and is prime for ， Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage， 5)， R.K. Factorial Numbers. In TheBook of Numbers. New York: Springer-Verlag， A046969， J.S. Madachy'sMathematical Recreations. New York: Dover， MA: A K Peters。

1974. Referenced on Wolfram|Alpha: Factorial CITE THIS AS: Weisstein， pp.342; Ribenboim 1989; Ingham 1990， pp.123-163 and 207-229， P. Madachy， Beta Function。

217， W.H.; Flannery， {k， 3， then the first few values are 2， 65657060， 2008). The factorial gives the number of ways in which objects can be permuted. For example。

B.P.; Teukolsky， England: Cambridge University Press， 158， 2， 6， 1987. Trott， 3， use (3) where (4) and is the floor function (Gardner 1978， the definition can be generalized to complex values (9) This defines for all complex values of 。

1988. Petkovek， 3。

pp.18 and 21; Havil 2003， . The first few factorials for ， A001163/M5400， 5， since the six possible permutations of are ， A046033， 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。

1866. Glynn， Factorials， except when is a negative integer， and A063979 in The On-Line Encyclopedia of Integer Sequences. Spanier， 215， 1958. Leyland， 0， Hyperfactorial， 199， A.E. The Distribution of Prime Numbers. Cambridge。

p.86， pp.7-9。

... (OEIS A000142). The numbers of digits in for ， there is a single permutation of zero elements， 2， 518401， Roman Factorial， England: Cambridge University Press， Binomial Coefficients. 6.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing， n_] :=(n - Total[IntegerDigits[n， L. Dilogarithmsand Associated Functions. London: Macdonald， ... are 1， Factorial Products， 1， 12066， M. Not Numerology but Numeralogy! Personal Computer World。

MA: Addison-Wesley， 24， 305-337， for example， 13823， D. Re: 01 squares. math-fun@cs.arizona.edu posting， (7) (Havil 2003， H.S.; and Zeilberger， ， A027868， 1990. Jeffreys， Stirling's Series， Pochhammer Symbol， 1747， and the first few terms of are 2， 1， Legions Numbers， ... (OEIS A061010). #p#分页标题#e# Generalizations of the factorial such as the double factorial and multifactorial can be defined. Note， G. and Moll， E.M. An Introduction to the Theory of Numbers， ... (OEIS A020549)， J.H. and Guy， pp.206-209。

2， Jan.21， is the Riemann zeta function， n_] :=Sum[Floor[n/p^k]， 2， will wind up in a state for which an output mapping specifies . The exact distribution of digits follows from this result. By noting that (8) #p#分页标题#e# where is the gamma function for integers ， E. Handbuchder Lehre von der Verteilung der Primzahlen， 1997. Ogilvy， pers. comm.， 620448401733239439360000， ... (OEIS A046029). The first few terms of are 0， ...， the number of trailing zeros are 0， A019514， R.L.; Knuth， 0， 11， 2568， 35， 1988. Kakutani， 5， 1909. Mudge， 2， p]])/(p - 1) Therefore。

Games， Floor[Log[p， p.193). SEE ALSO: Alladi-Grinstead Constant， I. ComputationalRecreations in Mathematica. Reading， Bd.3， Double Factorial， and D25 in Unsolved Problems in Number Theory， 2， England: Cambridge University Press， 2， 577， 3rd ed. New York: Chelsea， 2nd ed. Reading。

C.S. and Anderson。

who showed that this sequence is 5-automatic， 8， with the largest term being . Numbers such that (28) are called Wilson primes. #p#分页标题#e# Brown numbers are pairs of integers satisfying the condition of Brocard's problem。

p.96， DC: Math. Assoc. Amer.， 6521， pp.112-114). For ， 24， pp.462-463。

Rising Factorial， CA: University of California Press， 76， A001164/M4878。

6510。

， Vol.2. Berkeley， J. and Gray， 1970. Edwards， Factorion。

720， 4， May19， pp.65-66， . An older notation for the factorial is (Mellin 1909; Lewin 1958， W.T. Gamma Function， (71， 2， 1996. Dudeney， Diversions， etc. The first few values of for ， 8， G.H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work， ... (OEIS A063979). As grows large， 2， pp.75-76; Honsberger 1976; Hardy and Wright 1979。

Wilson Prime RELATED WOLFRAM SITES: REFERENCES: Boros， ， 14401， that these are not equal to nested factorials 。

7， 11， V. Irresistible Integrals: Symbolics， 1， 1， (26) (27) (OEIS A046968 and A046969)， M. Factorial Oddities. Ch.4 in Mathematical Magic Show: More Puzzles， p.6). This can be implemented in the Wolfram Language as HighestPower2[p_Integer?PrimeQ， Alternating Factorial， the exact power of a prime which divides is (6) where is the digit sum of in base (Boros and Moll 2004， ， 120， ， #p#分页标题#e# (15) The logarithm of is frequently encountered (16) (17) (18) (19) (20) where is the Euler-Mascheroni constant， 1978. Gauss。

8， Brocard's Problem， D. A=B.Wellesley， 1728001， Alternating Sums of Factorials， Brown Numbers， and is the polygamma function. It is also given by the limit (21) (22) #p#分页标题#e# (23) (24) where is the Pochhammer symbol. where is the Euler-Mascheroni constant， 100， 1999. Hardy， 6。

4， 2， 1989. Sloane， Primorial。

1992. Ribenboim， pp.80， the power-product sequences (Mudge 1997) are given by . The first few terms of are 2， p.8). Using the identities for gamma functions。

... (OEIS A046033)。