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 A008904 a(n) is the final nonzero digit of n!. 29
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. - Jean-Paul Allouche, Jul 25 2001 The decimal number 0.1126422428... 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, Feb 21 2006 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) REFERENCES J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 202. 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 S. Kakutani, Ergodic theory of shift transformations, in Proc. 5th Berkeley Symp. Math. Stat. Prob., Univ. Calif. Press, vol. II, 1967, 405-414. Popular Computing (Calabasas, CA), Problem 120, Factorials, Vol. 4 (No. 36, Mar 1976), page PC36-3. LINKS Robert G. Wilson v, Table of n, a(n) for n = 0..10000 K. S. Brown, The least significant nonzero digit of n! F. M. Dekking, Regularity and irregularity of sequences generated by automata Sém. Théor. Nombres, Bordeaux, Exposé 9, 1979-1980, pages 9-01 to 9-10. Jean-Marc Deshouillers, A footnote to the least non zero digit of n! in base 12, Uniform Distribution Theory 7:1 (2012), pp. 71-73. Jean-Marc Deshouillers, Yet Another Footnote to the Least Non Zero Digit of n! in Base 12. Unif. Distrib. Theory 11 (2016), no. 2, 163-167. 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. Gregory P. Dresden, Two Irrational Numbers from the Last Nonzero Digits of n! and n^n, Mathematics Magazine, Vol. 74, No. 4 (2001), 316-320. Gregory P. Dresden, Research Papers. Fritz Jacob (fritzjacob(AT)gmail.com), A way to compute a(n) MathPages, Least Significant Non-Zero Digit of n! J. C. Martin, The structure of generalized Morse minimal sets on m symbols, Trans. Amer. Math. Soc. 232 (1977), 343-355. T. Sillke, What are the next entries for the following sequences? Puzzle U asks for the next number after 2642242888682886824484644846. 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 FORMULA 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*Product_{i=0..N} (a_i)! (2^(i a_i)) mod 10. - Greg Dresden, 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)). - Reinhard Zumkeller, Aug 16 2008 From Washington Bomfim, Jan 09 2011: (Start) 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) = 2*a(n) mod 10, and    iii) a(n+4) = 4*a(n) mod 10. For k not equal to 1, a(10^k) = a(2^k). See second Dresden link, and second Bomfim link. (End) EXAMPLE 6! = 720, so a(6) = 2. MATHEMATICA 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[], #2!2^(#1[]#2)} &, {0, 0}, Reverse[IntegerDigits[n, 5]]]]), 10][]; Join[{1, 1}, Table[f[n], {n, 2, 100}]] (* program contributed by Jacob A. Siehler, Greg Dresden, 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}]] (* Harvey P. Dale, Dec 16 2010 *) f[n_]:=Block[{id=IntegerDigits[n!, 10]}, While[id[[-1]]==0, id=Most@id]; id[[-1]]]; Table[f@n, {n, 0, 100}] (* Vincenzo Librandi, Sep 07 2017 *) PROG (Python) 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 # 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) \\ Charles R Greathouse IV, Nov 05 2010; cleaned up by Max Alekseyev, Jan 28 2012 (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 CROSSREFS Cf. A008905, A000142, A004154, A034886. Indices of 2,4,6,8: A045547, A045548, A045549, A045550. Other bases: A136690, A136691, A136692, A136693, A136694, A136695, A136696, A136697, A136698, A136699, A136700, A136701, A136702. Sequence in context: A004600 A325497 A021795 * A334397 A074382 A061350 Adjacent sequences:  A008901 A008902 A008903 * A008905 A008906 A008907 KEYWORD nonn,base,nice AUTHOR EXTENSIONS More terms from Greg Dresden, Feb 21 2006 STATUS approved

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Last modified October 2 18:53 EDT 2022. Contains 357228 sequences. (Running on oeis4.)