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

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

Offset: 0

Author

Robert G. Wilson v, Jan 01 2011

Keywords

• nonn
• base
• more

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

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

Crossrefs

Cf. A008904, A173670

Programs

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}]

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

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)

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