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A008904
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Final nonzero digit of n!.
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21
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1, 1, 2, 6, 4, 2, 2, 4, 2, 8, 8, 8, 6, 8, 2, 8, 8, 6, 8, 2, 4, 4, 8, 4, 6, 4, 4, 8, 4, 6, 8, 8, 6, 8, 2, 2, 2, 4, 2, 8, 2, 2, 4, 2, 8, 6, 6, 2, 6, 4, 2, 2, 4, 2, 8, 4, 4, 8, 4, 6, 6, 6, 2, 6, 4, 6, 6, 2, 6, 4, 8, 8, 6, 8, 2, 4, 4, 8, 4, 6, 8, 8, 6, 8, 2, 2, 2, 4, 2, 8, 2, 2, 4, 2, 8, 6, 6, 2, 6
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OFFSET
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0,3
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COMMENTS
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Jean-Paul Allouche, Jul 25, 2001: this sequence is not ultimately periodic. This can be deduced from the fact that the sequence can be obtained as a fixed point of a morphism.
The decimal number .1126422428... formed from these digits is a transcendental number; see the article by G. Dresden. The Mathematica code uses Dresden's formula for the last nonzero digit of n!; this is more efficient than simply calculating n! and then taking its least-significant digit. - Greg Dresden (dresdeng(AT)wlu.edu), Feb 21 2006
Contribution from Robert G. Wilson v, Feb 16 2011: (Start)
.(mod 10) == 2..........4..........6..........8
10^
...1.........4..........2..........1..........1
...2........28.........23.........22.........25
...3.......248........247........260........243
...4......2509.......2486.......2494.......2509
...5.....25026......24999......24972......25001
...6....249993.....250012.....250040.....249953
...7...2500003....2499972....2499945....2500078
...8..25000078...24999872...25000045...25000003
...9.249999807..250000018..250000466..249999707 (End)
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 202.
F. M. Dekking, Regularity and irregularity of sequences generated by automata, S\'em. Th\'eor. Nombres, Bordeaux, Expos\'e 9, 1979-1980, pages 9-01 to 9-10.
J.-M. Deshouillers, A footnote to the least non zero digit of n! in base 12, Uniform Distribution Theory 7 (2012), no. 1, 71-73; http://www.boku.ac.at/math/udt/vol07/no1/03Desh30-11.pdf. - From N. J. A. Sloane, Oct 03 2012
Gregory P. Dresden, Three transcendental numbers from the last non-zero digits of n^n, F_n and n!, Mathematics Magazine, pp. 96-105, vol. 81, 2008.
S. Kakutani, Ergodic theory of shift transformations, in Proc. 5th Berkeley Symp. Math. Stat. Prob., Univ. Calif. Press, vol. II, 1967, 405-414.
J. C. Martin, The structure of generalized Morse minimal sets on m symbols, Trans. Amer. Math. Soc. 232 (1977), 343-355.
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LINKS
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_Robert G. Wilson v_, Table of n, a(n) for n = 0..10000 . [From Robert G. Wilson v, May 20 2010]
W. Bomfim, An algorithm to find the last nonzero digit of n!
W. Bomfim, A property of the last non-zero digit of factorials
K. S. Brown, The least significant nonzero digit of n!
Gregory P. Dresden, Two Irrational Numbers ...
G. Dresden, Home page.
Fritz Jacob (fritzjacob(AT)gmail.com), A way to compute a(n)
MathPages, Least Significant Non-Zero Digit of n!
Eric Weisstein's World of Mathematics, Factorial
David W. Wilson, Minimal state machine for this sequence
David W. Wilson, Another method for computing this sequence
Index entries for sequences related to final digits of numbers
Index entries for sequences related to factorial numbers
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FORMULA
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The generating function for n>1 is as follows: for n = a_0 + 5 a_1 + 5^2 a_2 + ... +5^N a_N (the expansion of n in base-5), then the last nonzero digit of n!, for n>1, is 6*\prod_{i=0}^N (a_i)! (2^(i a_i)) mod 10 - Greg Dresden (dresdeng(AT)wlu.edu), Feb 21 2006
a(n) = f(n,1,0) with f(n,x,e) = if n<2 then A010879(x*A000079(e)) else f(n-1,A010879(x*A132740(n),e+A007814(n)-A112765(n)). [From Reinhard Zumkeller, Aug 16 2008]
a(0) = 1, a(1) = 1, if n >= 2, with
n represented in base 5 as (a_h, ... ,a_1,a_0)5,
t = sum{i = h, h-1, ... , 0} (a_i even),
x = sum{i=h, h-1, ... , 1}(sum{k=h, h-1, ... , i}(a_i)),
z = (x + t/2) mod 4, and y = 2^z,
a(n) = 6(y mod 2) + y(1-(y mod 2)).
For n >= 5, and n mod 5 = 0, i) a(n) = a(n+1) = a(n+3), ii) a(n+2)=2a(n) mod 10, and iii) a(n+4)=4a(n) mod 10.
For k not equal to 1, a(10^k) = a(2^k). See second Dresden link, and second Bomfim link.
[W. Bomfim webonfim(AT)bol.com Jan 09, 2011]
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EXAMPLE
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6! = 720, so a(6) = 2.
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MATHEMATICA
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f[n_]:=Module[{m=n!}, While[Mod[m, 10]==0, m=m/10]; Mod[m, 10]]
Table[f[i], {i, 0, 100}]
f[n_] := Mod[6Times @@ (Rest[FoldList[{ 1 + #1[[1]], #2!2^(#1[[1]]#2)} &, {0, 0}, Reverse[IntegerDigits[n, 5]]]]), 10][[2]]; Join[{1, 1}, Table[f[n], {n, 2, 100}]] (* program contributed by Jacob A. Siehler *) - Greg Dresden (dresdeng(AT)wlu.edu), Feb 21 2006
zOF[n_Integer?Positive] := Module[{maxpow=0}, While[5^maxpow<=n, maxpow++]; Plus@@Table[Quotient[n, 5^i], {i, maxpow-1}]]; Flatten[Table[Take[IntegerDigits[n!], {-zOF[n]-1}], {n, 100}]] [From Harvey P. Dale, Dec. 16, 2010]
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PROG
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(Python) # replace triple dots by spaces
def a(n):
...if n<=1: return 1
...return 6*[1, 1, 2, 6, 4, 4, 4, 8, 4, 6][n%10]*3**(n/5%4)*a(n/5)%10
[From Maciej Ireneusz Wilczynski, Aug 23 2010]
(PARI) { a(n) = r=1; while(n>0, r *= Mod(4, 10)^((n\10)%2) * [1, 2, 6, 4, 2, 2, 4, 2, 8][max(n%10, 1)]; n\=5); lift(r) }
(Sage)
def A008904(n):
....# algorithm from David Wilson, http://oeis.org/A008904/a008904b.txt
....if n == 0 or n == 1: return 1
....dd = n.digits(base=5)
....x = sum(i*d for i, d in enumerate(dd))
....y = sum(d for d in dd if d % 2 == 0)/2
....z = 2**((x+y) % 4)
....if z == 1: z = 6
....return z # [D. S. McNeil, Dec 09 2010] (Haskell)
a008904 n = a008904_list !! n
a008904_list = 1 : 1 : f 2 1 where
f n x = x' `mod` 10 : f (n+1) x' where
x' = g (n * x) where
g m | m `mod` 5 > 0 = m
| otherwise = g (m `div` 10)
-- Reinhard Zumkeller, Apr 08 2011
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CROSSREFS
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Cf. A008905, A000142.
Sequence in context: A059574 A004600 A021795 * A074382 A061350 A046276
Adjacent sequences: A008901 A008902 A008903 * A008905 A008906 A008907
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KEYWORD
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nonn,base,nice
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AUTHOR
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Russ Cox
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EXTENSIONS
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More terms from Greg Dresden (dresdeng(AT)wlu.edu), Feb 21 2006
Pari code from Charles R Greathouse IV, Nov 05 2010
Pari code cleaned up by Max Alekseyev, Jan 28 2012
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STATUS
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approved
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