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 A173670 Last nonzero decimal digit of (10^n)!. 5

%I

%S 1,8,4,2,8,6,4,8,6,4,2,8,6,6,6,6,8,2,6,8,8,2,4,2,2,8,2,6,2,6,4,4,6,6,

%T 4,2,8,2,6,4,6,4,2,4,4,2,8,8,4,4,2,6,6,4,4,8,8,4,6,2,2,4,4,2,4,6,2,4,

%U 4,4,2,2,6,8,6,6,4,2,2,4,4,2,8,8,2,6,2,6,2,2,6,2,2,8,6,2,2,4,6,6

%N Last nonzero decimal digit of (10^n)!.

%C Except for n = 1, a(n) is also the last nonzero digit of(2^n)!. See the third Bomfim link. - _Washington Bomfim_, Jan 04 2011

%H W. Bomfim, <a href="http://oeis.org/w/images/7/77/BfileA173670.txt">Table of n, a(n) for n = 0..1000</a>

%H W. Bomfim, <a href="http://oeis.org/w/images/4/48/AlgLastFinal1.txt">An algorithm to find the last nonzero digit of n!</a>

%H W. Bomfim, <a href="http://oeis.org/w/images/6/61/Proof3.txt">A property of the last non-zero digit of factorials</a>

%F From _Washington Bomfim_, Jan 04 2011: (Start)

%F a(n) = A008904(10^n).

%F a(0) = 1, a(1) = 8, if n >= 2, with

%F 2^n represented in base 5 as (a_h, ... ,a_1,a_0)5,

%F t = sum{i = h, h-1, ... , 0} (a_i even),

%F x = sum{i=h, h-1, ... , 1}(sum{k=h, h-1, ... , i}(a_i)),

%F z = (x + t/2) mod 4, and y = 2^z,

%F a(n) = 6(y mod 2) + y(1-(y mod 2)).

%F (End)

%e a(1) = 8, because (10^1)! = 3628800

%t f[n_] := Mod[6Times @@ (Rest[FoldList[{ 1 + #1[[1]], #2!2^(#1[[1]]#2)} &, {0, 0}, Reverse[IntegerDigits[n, 5]]]]), 10][[2]]; f[0] = 1; Table[ f[10^n], {n, 0, 104}]] (* _Jacob A. Siehler_ *)

%o (Sage) A173670 = lambda n: A008904(10**n) # _D. S. McNeil_, Dec 14 2010

%o (PARI)\\ L is the list of the N digits of 2^n in base 5. \\ L[1] = a_0 ,..., L[N] = a_(N-1).

%o convert(n)={n=2^n; x=n; N=floor(log(n)/log(5)) + 1;

%o L = listcreate(N);

%o while(x, n=floor(n/5); r=x-5*n; listput(L, r); x=n;);

%o L; N

%o };

%o print("0 1");print("1 8");for(n=2,1000,print1(n," "); convert(n); q=0;t=0;x=0;forstep(i=N,2,-1,a_i=L[i];q+=a_i;x+=q;t+=a_i*(1-a_i%2););a_i=L[1];t+=a_i*(1-a_i%2);z=(x+t/2)%4;y=2^z;an=6*(y%2)+y*(1-(y%2)); print(an)); \\ _Washington Bomfim_, Dec 31 2010

%Y Cf. A008904, final nonzero digit of n!.

%Y Cf. A055476, Powers of ten written in base 5.

%Y Cf. A053824, Sum of digits of n written in base 5.

%K nonn,easy,base

%O 0,2

%A _Vladimir Reshetnikov_, Nov 24 2010

%E Extended by _D. S. McNeil_, Dec 12 2010

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Last modified April 13 07:16 EDT 2021. Contains 342935 sequences. (Running on oeis4.)