A008904 -id:A008904 - cp's OEIS frontend

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

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, 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, 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

Offset: 0

Vladimir Reshetnikov, Nov 24 2010

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

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

Washington Bomfim, Table of n, a(n) for n = 0..1000
Washington Bomfim, An algorithm to find the last nonzero digit of n!.
Washington Bomfim, A property of the last non-zero digit of factorials.

Cf. A008904, final nonzero digit of n!.
Cf. A055476, Powers of ten written in base 5.
Cf. A053824, Sum of digits of n written in base 5.

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

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

from functools import reduce
from sympy.ntheory.factor_ import digits
def A173670(n): return reduce(lambda x,y:x*y%10,((1,1,2,6,4)[a]*((6,2,4,8)[i*a&3] if i*a else 1) for i, a in enumerate(digits(1<Chai Wah Wu, Dec 07 2023

A173670 = lambda n: A008904(10**n)  # D. S. McNeil, Dec 14 2010

From Washington Bomfim, Jan 04 2011: (Start)
a(n) = A008904(10^n).
a(0) = 1, a(1) = 8, if n >= 2, with
2^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)). (End)

Extended by D. S. McNeil, Dec 12 2010

A073095 Numbers k such that the final nonzero digit of k! is the same as the last digit of binomial(2k,k) (in base 10).

Original entry on oeis.org

5, 12, 26, 31, 35, 51, 56, 60, 136, 152, 157, 177, 182, 252, 257, 275, 280, 287, 300, 305, 312, 627, 632, 650, 655, 662, 675, 680, 687, 751, 756, 760, 786, 811, 886, 902, 907, 927, 932, 1251, 1256, 1260, 1286, 1311, 1377, 1382, 1400, 1405, 1412, 1425

Offset: 1

Benoit Cloitre, Aug 18 2002

			12! = 479001600, binomial(24,12) = 2704156, and the last nonzero digit of 12! is the same as the last digit of binomial(24,12), hence 12 is in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Select[Range[1500],Mod[#!/10^IntegerExponent[#!,10],10]==Mod[Binomial[2 #,#],10]&] (* Harvey P. Dale, Sep 13 2022 *)

from math import comb
from functools import reduce
from itertools import count, zip_longest, islice
from sympy.ntheory.factor_ import digits
from sympy.ntheory.modular import crt
def A073095_gen(startvalue=2): # generator of terms >= startvalue
    for n in count(max(startvalue,2)):
        s, s2 = digits(n,5)[-1:0:-1], digits(n<<1,5)[-1:0:-1]
        if 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(s)),6)==crt([2,5],[0,reduce(lambda x,y:x*y%5,(comb(a, b) for a, b in zip_longest(s2,s,fillvalue=0)))])[0]:
            yield n
A073095_list = list(islice(A073095_gen(),50)) # Chai Wah Wu, Dec 07 2023

k such that A008904(k) = binomial(2k, k) reduced (mod 10).

A116081 Final nonzero digit of n^n.

Original entry on oeis.org

1, 4, 7, 6, 5, 6, 3, 6, 9, 1, 1, 6, 3, 6, 5, 6, 7, 4, 9, 6, 1, 4, 7, 6, 5, 6, 3, 6, 9, 9, 1, 6, 3, 6, 5, 6, 7, 4, 9, 6, 1, 4, 7, 6, 5, 6, 3, 6, 9, 5, 1, 6, 3, 6, 5, 6, 7, 4, 9, 6, 1, 4, 7, 6, 5, 6, 3, 6, 9, 9, 1, 6, 3, 6, 5, 6, 7, 4, 9, 6, 1, 4, 7, 6, 5, 6, 3, 6, 9, 1, 1, 6, 3, 6, 5, 6, 7, 4, 9, 1, 1, 4, 7, 6, 5

Offset: 1

Greg Dresden, Mar 12 2006

The decimal number .147656369116... formed from these digits is a transcendental number; see Dresden's second article. These digits are never eventually periodic.

Digits appear with predictable frequencies: 1/10 for 3, 4, and 7; 1/9 for 5; 3/25 for 9; 28/225 for 1; and 307/900 for 6. - Charles R Greathouse IV, Oct 03 2022

			a(4) = 6 because 4^4 (which is 256) ends in 6.

T. D. Noe, Table of n, a(n) for n = 1..1000
Articles can be found on Dresden's Home Page.
Gregory P. Dresden, Two Irrational Numbers From the Last Non-Zero Digits of n! and n^n, Math. Mag. 74 (October 2001), 316-320.
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.
Jose María Grau and A. M. Oller-Marcen, On the last digit and the last non-zero digit of n^n in base b., arXiv:1203.4066 [math.NT], 2012.
Jose María Grau and A. M. Oller-Marcen, On the last digit and the last non-zero digit of n^n in base b, Bull. Korean Math. Soc. 51 (2014), No. 5, pp. 1325-1337.
S. Ikeda and K. Matsuoka, On transcendental numbers generated by certain integer sequences, Siauliai Math. Semin. 8 (2013), 63-69. Mentions this sequence.
Index entries for transcendental numbers

Cf. A008904, A010879, A065881.
Cf. A056849.
Cf. A204815, A204816, A204817, A204818, A204819, A230024, A204697.
Cf. A322489, A322490.

f:= proc(n) local d, m, p; d:= min(padic:-ordp(n,2), padic:-ordp(n,5));
   m:= n/10^d;
   p:= n - 1 mod 4 + 1;
   m &^ p mod 10;
end proc:
seq(f(n), n=1..1000); # Robert Israel, Oct 19 2014

f[n_] := Block[{m = n}, While[ Mod[m, 10] == 0, m /= 10]; PowerMod[m, n, 10]]; Array[f, 105] (* Robert G. Wilson v, Mar 13 2006 and modified Oct 12 2014 *)

f(n) = while(!(n % 10), n/=10); n % 10; \\ A065881
a(n) = lift(Mod(f(n), 10)^n); \\ Michel Marcus, Sep 13 2022

a(n)=my(k=n/10^valuation(n,10)); lift(Mod(k,10)^(n%4+4)) \\ Charles R Greathouse IV, Sep 13 2022

def a(n):
    k = n
    while k%10 == 0: k //= 10
    return pow(k, n, 10)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Sep 13 2022

def A116081(n): return pow(int(str(n).rstrip('0')[-1]),n,10) # Chai Wah Wu, Dec 07 2023

a(n) = A065881(n)^n mod 10 = A010879(A065881(n)^(A010883(n-1))). - Robert Israel, Oct 19 2014

More terms from Robert G. Wilson v, Mar 13 2006

A208279 Central terms of Pascal's triangle mod 10 (A008975).

Original entry on oeis.org

1, 2, 6, 0, 0, 2, 4, 2, 0, 0, 6, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 4, 8, 4, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 2, 6, 0, 0, 2, 4, 2, 0, 0, 6, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Feb 25 2012

From Chai Wah Wu, Dec 08 2023: (Start)
Last digit of central binomial coefficient binomial(2n,n) in base 10.
A000984 mod 10.
A073095 are numbers n such that a(n) = A008904(n).
a(n) is even for n>0. (End)
From Robert Israel, Dec 08 2023: (Start)
If at least one base-5 digit of n is 3 or 4, then a(n) = 0.
Otherwise, if k 1's occur in the base-5 expansion of n, then a(n) = 2^k (mod 10) if k > 0, or 6 if k = 0. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A000984, A008904, A008975, A073095.

Haskell
```
a208279 n = a008975 (2*n) n
```

f:= proc(n) local A,v;
   A:= convert(n,base,5);
   if select(`>=`,A,3) <> [] then return 0 fi;
   v:= numboccur(1,A);
   if v > 0 then 2^v mod 10
   else 6
   fi
end proc:
f(0):= 1:
map(f, [$0..200]); # Robert Israel, Dec 08 2023

Array[Mod[Binomial[2#,#],10]&,100,0] (* Paolo Xausa, Dec 09 2023 *)

from sympy.ntheory.factor_ import digits
def A208279(n):
    if n == 0: return 1
    s = digits(n,5)[1:]
    return 0 if any(x>2 for x in s) else ((6,2,4,8)[a&3] if (a:=s.count(1)) else 6) # Chai Wah Wu, Dec 08 2023

a(n) = A008975(2*n,n) = binomial(2n,n) mod 10.

A063944 Final nonzero digit of (n!)! (A000197).

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 6, 4, 6, 8, 2, 8, 8, 6, 4, 4, 6, 6, 8, 2, 6, 4, 4, 8, 2, 2, 6, 2, 2, 6, 4, 6, 2, 2, 8, 4, 6, 8, 2, 2, 2, 2, 8, 6, 6, 6, 2, 2, 6, 8, 4, 2, 2, 2, 8, 8, 4, 4, 2, 6, 8, 6, 4, 6, 6, 4, 8, 2, 2, 4, 4, 2, 8, 2, 8, 2, 4, 2, 8, 8, 6, 8, 2, 8, 4, 4, 6, 8, 8, 6, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 2, 6, 4, 2, 8, 2, 4, 2

Offset: 0

Jason Earls, Sep 01 2001

David W. Wilson, Table of n, a(n) for n = 0..10000
Index entries for sequences related to final digits of numbers

Cf. A000197, A008904.

for(n=0,22,m=n!!; while(Mod(m,10) == 0,m=m/10); print(Mod(m,10)))

from functools import reduce
from math import prod, factorial
from sympy.ntheory.factor_ import digits
def A063944(n): return reduce(lambda x,y:x*y%10,((1,1,2,6,4)[a]*((6,2,4,8)[i*a&3] if i*a else 1) for i, a in enumerate(digits(factorial(n),5)[-1:0:-1])))*6%10 if n>1 else 1 # Chai Wah Wu, Dec 07 2023

More terms from David W. Wilson, Sep 05 2001, who remarks that "I'll tell you, computing (107!)! took up some disk space!"

A367799 Ordinal transform of the final nonzero digit of the factorial numbers.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 3, 2, 4, 1, 2, 3, 2, 4, 5, 5, 6, 3, 7, 6, 3, 4, 8, 5, 4, 6, 7, 9, 8, 5, 10, 11, 6, 12, 7, 8, 9, 9, 10, 13, 11, 12, 10, 13, 14, 7, 8, 14, 9, 11, 15, 16, 12, 17, 15, 13, 14, 16, 15, 10, 11, 12, 18, 13, 16, 14, 15, 19, 16, 17, 17, 18, 17, 19, 20

Offset: 0

Alois P. Heinz, Dec 07 2023

n is the a(n)-th nonnegative integer producing value A008904(n).

			a(11) = 3 because 11! = 39916800 is the third factorial with final nonzero digit 8 after 9! = 362880 and 10! = 3628800. A008904(k) = 8 for k = 9, 10, 11, ... .

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A000142, A004154, A008904, A368010.

from functools import reduce
from itertools import count, islice
from collections import Counter
from sympy.ntheory.factor_ import digits
def A367799_gen(): # generator of terms
    c = Counter()
    for n in count(0):
        c[m:=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(digits(n,5)[-1:0:-1])),6) if n>1 else 1]+=1
        yield c[m]
A367799_list = list(islice(A367799_gen(),50)) # Chai Wah Wu, Dec 08 2023

Ordinal transform of A008904.

a(n) = |{ j in {0..n} : A008904(j) = A008904(n) }|.

A056129 Final nonzero digit of n-th primorial.

Original entry on oeis.org

1, 2, 6, 3, 1, 1, 3, 1, 9, 7, 3, 3, 1, 1, 3, 1, 3, 7, 7, 9, 9, 7, 3, 9, 1, 7, 7, 1, 7, 3, 9, 3, 3, 1, 9, 1, 1, 7, 1, 7, 1, 9, 9, 9, 7, 9, 1, 1, 3, 1, 9, 7, 3, 3, 3, 1, 3, 7, 7, 9, 9, 7, 1, 7, 7, 1, 7, 7, 9, 3, 7, 1, 9, 3, 9, 1, 3, 7, 9, 9, 1, 9, 9, 9, 7, 3, 9, 1, 7, 7, 1, 7, 3, 1, 1, 9, 7, 3, 3, 9, 9

Offset: 0

Robert G. Wilson v, Jul 28 2000

For n > 2, a(n) is in {1, 3, 7, 9}.

Index entries for sequences related to final digits of numbers

Cf. A002110, A008904.

Do[p = Product[Prime[m], {m, 1, n}]; While[Mod[p, 10] == 0, p = p/10]; Print[Mod[p, 10]], {n, 0, 100}]

a(n)=if(n<3, return([1,2,6][n+1])); my(m=Mod(3,10)); forprime(p=7,prime(n), m*=p); lift(m) \\ Charles R Greathouse IV, Dec 03 2024

a(n) = prime(n)#/10 mod 10 for n > 2. - Charles R Greathouse IV, Dec 03 2024

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

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Jan 01 2011

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

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

Cf. A008904, A173670

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}]

\\ 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

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

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)

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

A113621 Numbers k such that the representation of k^2 is a substring of that of k!, in base 10.

Original entry on oeis.org

1, 20, 29, 170, 176, 241, 3136, 9800, 20309, 20486, 53663, 73793, 94836, 200000

Offset: 1

Giovanni Resta, Jan 26 2006

Using one of the fast algorithms for computing the last nonzero digit of the factorial (A008904) it is easy to see that also 200000000 and 2*10^16 are terms.

			29^2 = 841 and 29! = 8(841)761993739701954543616000000.

Cf. A033180, A008904.

CF. A000142, A000290.

lst={}; Do[If[{}!= StringPosition[ToString[n! ], ToString[n^2]], AppendTo[lst, n]], {n, 10000}]; lst

Extended by Giovanni Resta, Apr 04 2014

A176786 Last nonzero digit of A000043(n)!.

Original entry on oeis.org

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

Offset: 1

Washington Bomfim, Dec 07 2010

The C program, from first link, is based on a new method, see second link. It was developed from a congruence found in the first reference "Concrete Mathematics". The function D() of this program implements the simple division algorithm found in "D. E. Knuth, The Art of Computer Programming, V.2." (second reference). Another approach can be to use Dresden's formula that can be found from the third link. One can use the function LastDigit() of the mentioned program to find the last nonzero digit of N! for very large values of N. The factorial of the 47th (known) Mersenne prime has approximately 10^12,978,195 digits.

Many other algorithms for the general problem of finding the last nonzero digit of a factorial are given in A008904. [D. S. McNeil, Dec 10 2010]

			a(1) = 6 since the first Mersenne prime is 3, and 3! = 6.

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math.; Addison-Wesley, section 4, exercises 40, and 54.
D. E. Knuth, The Art of Computer Programming, vol.2, section 4.3.1, exercise 16.

W. Bomfim, C program
W. Bomfim, An algorithm to find the last nonzero digit of N!
Gregory P. Dresden, Three transcendental numbers from the last non-zero digits of n^n, F_n and n!, Math. Mag., pp. 96-105, vol. 81, 2008. See Lemma 7.

Cf. A000043, A000668, A008904, A034886.

a(n) = A008904(2^A000043(n)-1) = A008904(A000668(n)).

Terms for n <= 40 confirmed by D. S. McNeil, Dec 08 2010
a(41)-a(47) from Max Alekseyev, Jan 31 2012, Mar 16 2015, Dec 01 2019
a(48) from Chai Wah Wu, Dec 07 2023

cp's OEIS Frontend

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

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A073095 Numbers k such that the final nonzero digit of k! is the same as the last digit of binomial(2k,k) (in base 10).

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A116081 Final nonzero digit of n^n.

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A208279 Central terms of Pascal's triangle mod 10 (A008975).

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A063944 Final nonzero digit of (n!)! (A000197).

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A367799 Ordinal transform of the final nonzero digit of the factorial numbers.

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A056129 Final nonzero digit of n-th primorial.

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