The factorial is defined for a Positive Integer as

(1) |

As grows large, factorials begin acquiring tails of trailing Zeros. To calculate the number of
trailing Zeros for , use

(2) |

(3) |

(4) |

(5) |

By noting that

(6) |

(7) |

(8) | |||

(9) | |||

(10) | |||

(11) |

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) |

(21) |

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 Bessel Function of the First Kind, cosh is the Hyperbolic Cosine, cos is the Cosine, sinh is the Hyperbolic Sine, and sin is the Sine.

Let be the exponent of the greatest Power of a Prime dividing . Then

(30) |

(31) |

(32) |

The sum-of-factorials function is defined by

(33) | |||

(34) |

where is the Exponential Integral, is the En-Function, and

(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 for only 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 called
Factorions. While no factorial is a Square Number, D. Hoey listed sums of distinct
factorials 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) |

(38) | |||

(39) | |||

(40) |

(Guy 1994, p. 80).

There are three numbers less than 200,000 for which

(41) |

(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. Sequences A014615, A014597, A033312, A020549, A000142/M1675, and A007489/M2818 in ``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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1999-05-26