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

8, 2, 6, 4, 2, 2, 6, 2, 6, 4, 2, 2

Offset

0,1

Comments

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

Links

Table of n, a(n) for n=0..11.

W. Bomfim, A property of the last non-zero digit of factorials.

W. Bomfim, C program.

Formula

From Washington Bomfim, Jan 06 2011: (Start)

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

a(n) = A008904(2^(10^n)), if n > 0.

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)).

(End)

Mathematica

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}]

Prog

(PARI)\\ L is the list of the N digits of 2^(10^n) in base 5.
\\ With 2^(10^n) in base 5 as (a_h, ... , a_0)5,
\\ L[1] = a_0, ... , L[N] = a_h.
convert(n)={n=2^(10^n); x=n; N=floor(log(n)/log(5)) + 1;
L = listcreate(N);
while(x, n=floor(n/5); r=x-5*n; listput(L, r); x=n; );
L; N
};
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
(Python)
from functools import reduce
from sympy.ntheory.factor_ import digits
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

Crossrefs

Cf. A008904, A173670

Sequence in context: A228818 A195845 A093207 * A019660 A021125 A329933

Adjacent sequences: A178966 A178967 A178968 * A178970 A178971 A178972

Keyword

nonn,base,more

Author

Robert G. Wilson v, Jan 01 2011

Extensions

a(9) from Washington Bomfim, Jan 06 2011

a(10) from Chai Wah Wu, Dec 07 2023

a(11) from Chai Wah Wu, Dec 08 2023

Status

approved