| 释义 |
FactorialThe factorial  is defined for a Positive Integer  as
 | (1) |
 , 1, 2, ... are 1, 1, 2, 6, 24, 120, ... (Sloane's A000142). An olderNotation for the factorial is  (Dudeney 1970, Gardner 1978, Conway and Guy1996).
As grows large, factorials begin acquiring tails of trailing Zeros. To calculate the number oftrailing Zeros for , use
 | (2) |
 | (3) |
 is the Floor Function (Gardner 1978, p. 63; Ogilvy and Anderson 1988, pp. 112-114). For  , 2,..., the number of trailing zeros are 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, ... (Sloane's A027868). This is aspecial application of the general result that the Power of a Prime  dividing is
 | (4) |
 | (5) |
By noting that
 | (6) |
 is the Gamma Function for Integers , the definition can be generalized toComplex values
 | (7) |
 for all Complex values of  , except when is a NegativeInteger, in which case  . Using the identities for Gamma Functions, thevalues of  (half integral values) can be written explicitly
where  is a Double Factorial.
For Integers  and with  ,
 | (12) |
 |  |  | (13) | | | |  | (14) | | | |  | (15) | | | |  | (16) | | | |  | (17) | | | |  | (18) | | | |  | (19) |
where  is the Euler-Mascheroni Constant,  is the Riemann Zeta Function, and  is the Polygamma Function. The factorial can be expanded in a series
 | (20) |
 ,
where  is a Bernoulli Number.
Identities satisfied by sums of factorials include
 | |  | (22) |  | |  | (23) |  | |  | (24) |  | |  | (25) |  | |  | (26) |  | |  | (27) |  | |  | (28) |  | |  | (29) |
(Spanier and Oldham 1987), where  is a Modified Bessel Function of the First Kind,  is a BesselFunction of the First Kind, cosh is the Hyperbolic Cosine, cos is the Cosine, sinh is the HyperbolicSine, and sin is the Sine.
Let  be the exponent of the greatest Power of a Prime dividing . Then
 | (30) |
 be the number of 1s in the Binary representation of . Then
 | (31) |
 of the Prime dividing is given by
 | (32) |
 , ...,  are the digits of in base (Ribenboim 1989).
The sum-of-factorials function is defined by
where  is the Exponential Integral,  is theEn-Function, and i is the Imaginary Number. The first few values are 1, 3, 9,33, 153, 873, 5913, 46233, 409113, ... (Sloane's A007489).  cannot be written as a hypergeometric term plus a constant (Petkovsek et al. 1996). However the sum
 | (35) |
The numbers  are prime for , 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, ... (Sloane's A002981), and the numbers  are prime for  , 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, ... (Sloane's A002982). In general, the power-product sequences(Mudge 1997) are given by  . The first few terms of  are 2, 5, 37, 577, 14401, 518401,... (Sloane's A020549), and is Prime for , 2, 3, 4, 5, 9, 10, 11, 13, 24, 65, 76, ... (Sloane's A046029). The first few terms of  are 0, 3, 35, 575, 14399, 518399, ... (Sloane's A046032), but is Prime foronly  since  for  . The first few terms of  are 0, 7, 215, 13823, 1727999,... (Sloane's A0460333), and the first few terms of  are 2, 9, 217, 13825, 1728001, ... (Sloane's A019514).
There are only four Integers equal to the sum of the factorials of their digits. Such numbers are calledFactorions. While no factorial is a Square Number, D. Hoey listed sums  of distinctfactorials which give Square Numbers, and J. McCranie gave the one additional sum less than :
(Sloane's A014597). The first few values for which the alternating Sum
 | (36) |
(Madachy 1979).
There are no identities of the form
 | (37) |
 with  for  for  except
(Guy 1994, p. 80).
There are three numbers less than 200,000 for which
 | (41) |
 of Integers satisfyingthe condition of Brocard's Problem, i.e., such that
 | (42) |
References
Conway, J. H. and Guy, R. K. ``Factorial Numbers.'' In The Book of Numbers. New York: Springer-Verlag, pp. 65-66, 1996.Dudeney, H. E. Amusements in Mathematics. New York: Dover, p. 96, 1970. Gardner, M. ``Factorial Oddities.'' Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 50-65, 1978. Graham, R. L.; Knuth, D. E.; and Patashnik, O. ``Factorial Factors.'' §4.4 in Concrete Mathematics: A Foundation for Computer Science. Reading, MA: Addison-Wesley, pp. 111--115, 1990. Guy, R. K. ``Equal Products of Factorials,'' ``Alternating Sums of Factorials,'' and ``Equations Involving Factorial .'' §B23, B43, and D25 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 80, 100, and 193-194, 1994. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 2, 1976. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983. Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/factorial-.Z and ftp://sable.ox.ac.uk/pub/math/factors/factorial+.Z. Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 174, 1979. Mudge, M. ``Not Numerology but Numeralogy!'' Personal Computer World, 279-280, 1997. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, 1988. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, p. 86, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Gamma Function, Beta Function, Factorials, Binomial Coefficients.'' §6.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 206-209, 1992. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 22-24, 1989. Sloane, N. J. A. SequencesA014615,A014597,A033312,A020549,A000142/M1675, andA007489/M2818in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html. Spanier, J. and Oldham, K. B. ``The Factorial Function and Its Reciprocal.'' Ch. 2 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 19-33, 1987. Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 67, 1991.
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