cp's OEIS Frontend

A178969 Last nonzero decimal digit of (10^10^n)!.

%I A178969 #39 Dec 08 2023 09:36:31
%S A178969 8,2,6,4,2,2,6,2,6,4,2,2
%N A178969 Last nonzero decimal digit of (10^10^n)!.
%C A178969 It is possible to find a(10) using the program from the second Bomfim link, or a similar GMP program. Reserve 312500001 words instead of 31250001. Use a computer with at least 6 GB of RAM. - _Washington Bomfim_, Jan 06 2011
%H A178969 W. Bomfim, <a href="http://oeis.org/w/images/6/61/Proof3.txt">A property of the last non-zero digit of factorials</a>.
%H A178969 W. Bomfim, <a href="http://oeis.org/w/images/c/cb/Program.txt">C program</a>.
%F A178969 From _Washington Bomfim_, Jan 06 2011: (Start)
%F A178969 a(n) = A008904(10^(10^n)).
%F A178969 a(n) = A008904(2^(10^n)), if n > 0.
%F A178969 For n >= 1, with N = 10^n, 2^N represented in base 5 as (a_h, ..., 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)).
%F A178969 (End)
%t A178969 f[n_] := Mod[6Times @@ (Rest[FoldList[{ 1 + #1[[1]], #2!2^(#1[[1]]#2)} &, {0, 0}, Reverse[IntegerDigits[n, 5]]]]), 10][[2]]; (* _Jacob A. Siehler_ *) Table[ f[10^10^n], {n, 0, 4}]
%o A178969 (PARI)\\ L is the list of the N digits of 2^(10^n) in base 5.
%o A178969 \\ With 2^(10^n) in base 5 as (a_h, ... , a_0)5,
%o A178969 \\ L[1] = a_0, ... ,L[N] = a_h.
%o A178969 convert(n)={n=2^(10^n); x=n; N=floor(log(n)/log(5)) + 1;
%o A178969 L = listcreate(N);
%o A178969 while(x, n=floor(n/5); r=x-5*n; listput(L, r); x=n; );
%o A178969 L; N
%o A178969 };
%o A178969 print("0 8"); for(n=1, 5, 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_, Jan 06 2011
%o A178969 (Python)
%o A178969 from functools import reduce
%o A178969 from sympy.ntheory.factor_ import digits
%o A178969 def A178969(n): return reduce(lambda x,y:x*y%10,(((6,2,4,8,6,2,4,8,2,4,8,6,6,2,4,8,4,8,6,2)[(a<<2)|(i*a&3)] if i*a else (1,1,2,6,4)[a]) for i, a in enumerate(sympydigits(1<<10**n,5)[-1:0:-1])),6) if n else 8 # _Chai Wah Wu_, Dec 07 2023
%Y A178969 Cf. A008904, A173670
%K A178969 nonn,base,more
%O A178969 0,1
%A A178969 _Robert G. Wilson v_, Jan 01 2011
%E A178969 a(9) from _Washington Bomfim_, Jan 06 2011
%E A178969 a(10) from _Chai Wah Wu_, Dec 07 2023
%E A178969 a(11) from _Chai Wah Wu_, Dec 08 2023