A136690 -id:A136690 - cp's OEIS frontend

A008904 a(n) is the final nonzero digit of n!.

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

Offset: 0

Russ Cox

This sequence is not ultimately periodic. This can be deduced from the fact that the sequence can be obtained as a fixed point of a morphism. - Jean-Paul Allouche, Jul 25 2001
The decimal number 0.1126422428... formed from these digits is a transcendental number; see the article by G. Dresden. The Mathematica code uses Dresden's formula for the last nonzero digit of n!; this is more efficient than simply calculating n! and then taking its least-significant digit. - Greg Dresden, Feb 21 2006
From Robert G. Wilson v, Feb 16 2011: (Start)
  (mod 10) == 2          4          6          8
10^
   1         4          2          1          1
   2        28         23         22         25
   3       248        247        260        243
   4      2509       2486       2494       2509
   5     25026      24999      24972      25001
   6    249993     250012     250040     249953
   7   2500003    2499972    2499945    2500078
   8  25000078   24999872   25000045   25000003
   9 249999807  250000018  250000466  249999707 (End)

			6! = 720, so a(6) = 2.

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 202.
Gardner, M. "Factorial Oddities." Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 50-65, 1978
S. Kakutani, Ergodic theory of shift transformations, in Proc. 5th Berkeley Symp. Math. Stat. Prob., Univ. Calif. Press, vol. II, 1967, 405-414.
Popular Computing (Calabasas, CA), Problem 120, Factorials, Vol. 4 (No. 36, Mar 1976), page PC36-3.

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
Washington Bomfim, An algorithm to find the last nonzero digit of n!
Washington Bomfim, A property of the last non-zero digit of factorials
K. S. Brown, The least significant nonzero digit of n!
F. M. Dekking, Regularity and irregularity of sequences generated by automata, Sém. Théor. Nombres, Bordeaux, Exposé 9, 1979-1980, pages 9-01 to 9-10.
Jean-Marc Deshouillers, A footnote to the least non zero digit of n! in base 12, Uniform Distribution Theory 7:1 (2012), pp. 71-73.
Jean-Marc Deshouillers, Yet Another Footnote to the Least Non Zero Digit of n! in Base 12. Unif. Distrib. Theory 11 (2016), no. 2, 163-167.
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.
Gregory P. Dresden, Two Irrational Numbers from the Last Nonzero Digits of n! and n^n, Mathematics Magazine, Vol. 74, No. 4 (2001), 316-320.
Gregory P. Dresden, Research Papers.
Fritz Jacob (fritzjacob(AT)gmail.com), A way to compute a(n)
MathPages, Least Significant Non-Zero Digit of n!
J. C. Martin, The structure of generalized Morse minimal sets on m symbols, Trans. Amer. Math. Soc. 232 (1977), 343-355.
T. Sillke, What are the next entries for the following sequences? Puzzle U asks for the next number after 2642242888682886824484644846.
Eric Weisstein's World of Mathematics, Factorial
David W. Wilson, Minimal state machine for this sequence
David W. Wilson, Another method for computing this sequence
Index entries for sequences related to final digits of numbers
Index entries for sequences related to factorial numbers

Cf. A008905, A000142, A004154, A034886.
Indices of 2,4,6,8: A045547, A045548, A045549, A045550.
Other bases: A136690, A136691, A136692, A136693, A136694, A136695, A136696, A136697, A136698, A136699, A136700, A136701, A136702.

a008904 n = a008904_list !! n
a008904_list = 1 : 1 : f 2 1 where
   f n x = x' `mod` 10 : f (n+1) x' where
      x' = g (n * x) where
         g m | m `mod` 5 > 0 = m
             | otherwise     = g (m `div` 10)
-- Reinhard Zumkeller, Apr 08 2011

f[n_]:=Module[{m=n!},While[Mod[m,10]==0,m=m/10];Mod[m,10]]
Table[f[i],{i,0,100}]
f[n_] := Mod[6Times @@ (Rest[FoldList[{ 1 + #1[[1]], #2!2^(#1[[1]]#2)} &, {0, 0}, Reverse[IntegerDigits[n, 5]]]]), 10][[2]]; Join[{1, 1}, Table[f[n], {n, 2, 100}]] (* program contributed by Jacob A. Siehler, Greg Dresden, Feb 21 2006 *)
zOF[n_Integer?Positive] := Module[{maxpow=0}, While[5^maxpow<=n,maxpow++]; Plus@@Table[Quotient[n,5^i], {i,maxpow-1}]]; Flatten[Table[ Take[ IntegerDigits[ n!], {-zOF[n]-1}],{n,100}]] (* Harvey P. Dale, Dec 16 2010 *)
f[n_]:=Block[{id=IntegerDigits[n!, 10]}, While[id[[-1]]==0, id=Most@id]; id[[-1]]]; Table[f@n, {n, 0, 100}] (* Vincenzo Librandi, Sep 07 2017 *)

a(n) = r=1; while(n>0, r *= Mod(4, 10)^((n\10)%2) * [1, 2, 6, 4, 2, 2, 4, 2, 8][max(n%10, 1)]; n\=5); lift(r) \\ Charles R Greathouse IV, Nov 05 2010; cleaned up by Max Alekseyev, Jan 28 2012

def a(n):
    if n <= 1: return 1
    return 6*[1,1,2,6,4,4,4,8,4,6][n%10]*3**(n/5%4)*a(n/5)%10
# Maciej Ireneusz Wilczynski, Aug 23 2010

from functools import reduce
from sympy.ntheory.factor_ import digits
def A008904(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(digits(n,5)[-1:0:-1])),6) if n>1 else 1 # Chai Wah Wu, Dec 07 2023

def A008904(n):
    # algorithm from David Wilson, http://oeis.org/A008904/a008904b.txt
    if n == 0 or n == 1: return 1
    dd = n.digits(base=5)
    x = sum(i*d for i,d in enumerate(dd))
    y = sum(d for d in dd if d % 2 == 0)/2
    z = 2**((x+y) % 4)
    if z == 1: z = 6
    return z # D. S. McNeil, Dec 09 2010

The generating function for n>1 is as follows: for n = a_0 + 5*a_1 + 5^2*a_2 + ... + 5^N*a_N (the expansion of n in base-5), then the last nonzero digit of n!, for n>1, is 6*Product_{i=0..N} (a_i)! (2^(i a_i)) mod 10. - Greg Dresden, Feb 21 2006
a(n) = f(n,1,0) with f(n,x,e) = if n < 2 then A010879(x*A000079(e)) else f(n-1, A010879(x)*A132740(n), e+A007814(n)-A112765(n)). - Reinhard Zumkeller, Aug 16 2008
From Washington Bomfim, Jan 09 2011: (Start)
a(0) = 1, a(1) = 1, if n >= 2, with
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)).
For n >= 5, and n mod 5 = 0,
     i) a(n) = a(n+1) = a(n+3),
    ii) a(n+2) = 2*a(n) mod 10, and
   iii) a(n+4) = 4*a(n) mod 10.
For k not equal to 1, a(10^k) = a(2^k). See second Dresden link, and second Bomfim link.
(End)

More terms from Greg Dresden, Feb 21 2006

A136691 Final nonzero digit of n! in base 4.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 1, 3, 2, 2, 3, 1, 3, 3, 2, 2, 2, 2, 3, 1, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 1, 3, 2, 2, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 3, 1, 3, 3, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 3, 2, 2, 2, 2, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 2, 2, 2, 1, 3, 2, 2, 1, 3, 1, 1, 2, 2, 3, 3, 2, 2

Offset: 0

Carl R. White, Jan 16 2008

			6! = 720 decimal = 23100 quartal, so a(6) = 1.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000142, A136690, A136692, A136693, A136694, A136695, A136696, A008904, A136697, A136698, A136699, A136700, A136701, A136702.

nzd[n_]:=Module[{rd=RealDigits[n!,4][[1]]},If[Last[rd]!=0,Last[rd], Last[ Flatten[Most[Split[rd]]]]]]; Array[nzd,100,0] (* Harvey P. Dale, May 08 2014 *)

A136695 Final nonzero digit of n! in base 8.

Original entry on oeis.org

1, 1, 2, 6, 3, 7, 2, 6, 6, 6, 4, 4, 6, 6, 4, 4, 3, 3, 6, 2, 5, 1, 6, 2, 6, 6, 4, 4, 6, 6, 4, 4, 6, 6, 4, 4, 2, 2, 4, 4, 4, 4, 1, 3, 4, 4, 3, 5, 6, 6, 4, 4, 2, 2, 4, 4, 4, 4, 3, 1, 4, 4, 5, 3, 3, 3, 6, 2, 1, 5, 6, 2, 2, 2, 4, 4, 2, 2, 4, 4, 5, 5, 2, 6, 3, 7, 2, 6, 2, 2, 4, 4, 2, 2, 4, 4, 6, 6, 4, 4

Offset: 0

Carl R. White, Jan 16 2008

			6! = 720 decimal = 1320 octal, so a(6) = 2.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000142, A136690, A136691, A136692, A136693, A136694, A136696, A008904, A136697, A136698, A136699, A136700, A136701, A136702.

Cf. A011371, A049606.

P:= 1: E:= 0: a[0]:= 1:
for n from 1 to 100 do
  v:= padic:-ordp(n,2);
  P:= P*(n/2^v) mod 8;
  E:= E + v;
  if E mod 3 = 0 then a[n]:= P
  elif E mod 3 = 1 then a[n]:= 2*(P mod 4)
  else a[n]:= 4
  fi
od:
seq(a[n],n=0..100); # Robert Israel, Sep 26 2018

Table[IntegerDigits[FromDigits[Reverse[IntegerDigits[n!,8]]]][[1]],{n,0,100}] (* Harvey P. Dale, Nov 29 2024 *)

From Robert Israel, Sep 26 2018: (Start)
If A011371(n) == 0 (mod 3) then a(n) = A049606(n) mod 8.
If A011371(n) == 1 (mod 3) then a(n) = 2*(A049606(n) mod 4).
If A011371(n) == 2 (mod 3) then a(n) = 4. (End)

A136698 Final nonzero digit of n! in base 12.

Original entry on oeis.org

1, 1, 2, 6, 2, 10, 5, 11, 4, 6, 7, 5, 5, 5, 10, 6, 8, 4, 11, 5, 4, 7, 10, 2, 4, 4, 8, 6, 6, 6, 3, 9, 8, 10, 4, 8, 11, 11, 10, 6, 4, 8, 10, 10, 8, 6, 1, 11, 8, 8, 4, 2, 8, 4, 9, 3, 10, 6, 1, 11, 7, 7, 2, 6, 8, 4, 4, 4, 8, 4, 4, 8, 4, 4, 8, 2, 8, 4, 8, 8, 4, 3, 6, 6, 6, 6, 7, 9, 2, 10, 3, 9, 5, 9, 6, 6, 8, 8, 4

Offset: 0

Carl R. White, Jan 16 2008

The asymptotic densities of the numbers k such that a(k) = m is 1/2 for m = 4 or 8, and 0 otherwise (Deshouillers and Ruzsa, 2011). Therefore, the asymptotic mean of this sequence is 6. There are infinitely many numbers k such that a(k) = m for each of the values m = 3, 6, or 9 (Deshouillers, 2012). - Amiram Eldar, Jan 11 2021

			6! = 720 decimal = 500 duodecimal, so a(6) = 5.

Michael De Vlieger, Table of n, a(n) for n = 0..10368 (10368 = 6 * 12^3)
Jean-Marc Deshouillers, A footnote to the least non zero digit of n! in base 12, Uniform Distribution Theory 7:1 (2012), pp. 71-73. [Wayback Machine link]
Jean-Marc Deshouillers, Yet Another Footnote to the Least Non Zero Digit of n! in Base 12, Unif. Distrib. Theory 11 (2016), no. 2, 163-167.
Jean-Marc Deshouillers and Imre Ruzsa, The least nonzero digit of n! in base 12, Publicationes Mathematicae, Vol. 79, No. 3-4 (2011), pp. 395-400.
Jean-Marc Deshouillers, Laurent Habsieger, Shanta Laishram, and Bernard Landreau, Sums of the digits in bases 2 and 3, arXiv:1611.08180 [math.NT], 2016. (See first page footnote.)
Jean-Marc Deshouillers, Pascal Jelinek, and Lukas Spiegelhofer, Binary-ternary collisions and the last significant digit of n! in base 12, arXiv:2412.09124 [math.NT], 2024.

Cf. A000142, A136690, A136691, A136692, A136693, A136694, A136695, A136696, A008904, A136697, A136699, A136700, A136701, A136702.

a136698[n_Integer] := Last[Select[IntegerDigits[n!, 12], # > 0 &]]; a136698 /@ Range[0, 144] (* Michael De Vlieger, Aug 13 2014 *)

a(n) = my(d=digits(n!, 12)); forstep(k=#d, 1, -1, if (d[k], return(d[k]))); \\ Michel Marcus, Feb 18 2025

A136692 Final nonzero digit of n! in base 5.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 4, 3, 4, 1, 2, 2, 4, 2, 3, 4, 4, 3, 4, 1, 4, 4, 3, 4, 1, 1, 1, 2, 1, 4, 4, 4, 3, 4, 1, 2, 2, 4, 2, 3, 4, 4, 3, 4, 1, 4, 4, 3, 4, 1, 2, 2, 4, 2, 3, 3, 3, 1, 3, 2, 4, 4, 3, 4, 1, 3, 3, 1, 3, 2, 3, 3, 1, 3, 2, 1, 1, 2, 1, 4, 4, 4, 3, 4, 1, 2, 2, 4, 2, 3, 4, 4, 3, 4, 1, 4, 4, 3, 4, 1

Offset: 0

Carl R. White, Jan 16 2008

			6! = 720 decimal = 10340 quinary, so a(6) = 4.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000142, A136690, A136691, A136693, A136694, A136695, A136696, A008904, A136697, A136698, A136699, A136700, A136701, A136702.

Cf. A243757.

f:= proc(n) option remember;
   local t;
   t:= n / 5^padic:-ordp(n,5);
   procname(n-1)*t mod 5
end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Jun 27 2025

Table[Module[{sp=Split[IntegerDigits[n!,5]]},If[sp[[-1,-1]]==0,sp[[-2, -1]], sp[[-1,-1]]]],{n,0,100}] (* Harvey P. Dale, Oct 26 2016 *)

a(n)={my(v=[1, 1, 2, 1, 4, 4, 4, 3, 4, 1]); if(n==0, 1, lift(prod(k=0, logint(n,5), Mod(v[1 + n\5^k % 10], 5))))} \\ Andrew Howroyd, Sep 27 2024

a(n) = (n!/A243757(n)) mod 5. - Ridouane Oudra, Sep 27 2024

A136694 Final nonzero digit of n! in base 7.

Original entry on oeis.org

1, 1, 2, 6, 3, 1, 6, 6, 6, 5, 1, 4, 6, 1, 2, 2, 4, 5, 6, 2, 5, 1, 1, 2, 6, 3, 1, 6, 3, 3, 6, 4, 2, 3, 4, 6, 6, 5, 1, 4, 6, 1, 6, 6, 5, 1, 4, 6, 1, 1, 1, 2, 6, 3, 1, 6, 6, 6, 5, 1, 4, 6, 1, 2, 2, 4, 5, 6, 2, 5, 1, 1, 2, 6, 3, 1, 6, 3, 3, 6, 4, 2, 3, 4, 6, 6, 5, 1, 4, 6, 1, 6, 6, 5, 1, 4, 6, 1, 2, 2

Offset: 0

Carl R. White, Jan 16 2008

			6! = 720 decimal = 2046 septimal, so a(6) = 6.

Cf. A000142, A136690, A136691, A136692, A136693, A136695, A136696, A008904, A136697, A136698, A136699, A136700, A136701, A136702.

A136696 Final nonzero digit of n! in base 9.

Original entry on oeis.org

1, 1, 2, 6, 6, 3, 8, 2, 7, 7, 7, 5, 6, 6, 3, 2, 5, 4, 8, 8, 7, 3, 3, 6, 1, 7, 2, 6, 6, 3, 4, 7, 8, 3, 3, 6, 6, 6, 3, 7, 1, 5, 3, 3, 6, 3, 3, 6, 8, 5, 7, 6, 6, 3, 8, 8, 7, 3, 3, 6, 1, 7, 2, 5, 5, 1, 3, 3, 6, 4, 1, 8, 1, 1, 2, 6, 6, 3, 8, 2, 7, 7, 7, 5, 6, 6, 3, 2, 5, 4, 4, 4, 8, 6, 6, 3, 5, 8, 1, 2

Offset: 0

Carl R. White, Jan 16 2008

			6! = 720 decimal = 880 nonary, so a(6) = 8.

Cf. A000142, A136690, A136691, A136692, A136693, A136694, A136695, A008904, A136697, A136698, A136699, A136700, A136701, A136702.

A136697 Final nonzero digit of n! in base 11.

Original entry on oeis.org

1, 1, 2, 6, 2, 10, 5, 2, 5, 1, 10, 10, 10, 9, 5, 9, 1, 6, 9, 6, 10, 1, 2, 2, 4, 1, 4, 9, 10, 4, 10, 2, 9, 5, 5, 10, 8, 10, 6, 3, 10, 3, 5, 6, 2, 2, 4, 1, 4, 9, 10, 4, 10, 2, 9, 1, 1, 2, 6, 2, 10, 5, 2, 5, 1, 10, 5, 5, 10, 8, 10, 6, 3, 10, 3, 5, 6, 9, 9, 7, 10, 7, 2, 1, 7, 1, 9, 2, 5, 5, 10, 8, 10, 6, 3

Offset: 0

Carl R. White, Jan 16 2008

			6! = 720 decimal = 5A5 undecimal, so a(6) = 5.

Cf. A000142, A136690, A136691, A136692, A136693, A136694, A136695, A136696, A008904, A136698, A136699, A136700, A136701, A136702.

A136699 Final nonzero digit of n! in base 13.

Original entry on oeis.org

1, 1, 2, 6, 11, 3, 5, 9, 7, 11, 6, 1, 12, 12, 12, 11, 7, 2, 10, 8, 4, 6, 2, 7, 12, 1, 2, 2, 4, 12, 9, 6, 10, 5, 1, 9, 12, 2, 11, 7, 7, 1, 3, 12, 8, 9, 11, 10, 12, 3, 7, 6, 11, 11, 9, 1, 4, 7, 3, 8, 12, 4, 1, 11, 2, 10, 10, 7, 8, 6, 4, 11, 12, 5, 6, 8, 10, 3, 5, 5, 10, 4, 3, 2, 12, 6, 9, 3, 4, 5, 8, 4

Offset: 0

Carl R. White, Jan 16 2008

			6! = 720 decimal = 435 tridecimal, so a(6) = 5.

Cf. A000142, A136690, A136691, A136692, A136693, A136694, A136695, A136696, A008904, A136697, A136698, A136700, A136701, A136702.

f[s_List] := Block[{a = s[[ -1]], len = Length@s}, Append[s, If[Mod[len, 13] == 0, Mod[a*len/13,13], Mod[a*len, 13]]]]; Nest[f, {1}, 100] (* Robert G. Wilson v, May 03 2009 *)
f[n_] := Block[{id = IntegerDigits[n!, 13]}, While[id[[ -1]] == 0, id = Most@id]; id[[ -1]]]; Table[ f@n, {n, 0, 100}] (* Robert G. Wilson v, May 03 2009 *)

A136701 Final nonzero digit of n! in base 15.

Original entry on oeis.org

1, 1, 2, 6, 9, 8, 3, 6, 3, 12, 3, 3, 6, 3, 12, 12, 12, 9, 12, 3, 9, 9, 3, 9, 6, 9, 9, 3, 9, 6, 12, 12, 9, 12, 3, 12, 12, 9, 12, 3, 3, 3, 6, 3, 12, 6, 6, 12, 6, 9, 12, 12, 9, 12, 3, 6, 6, 12, 6, 9, 6, 6, 12, 6, 9, 9, 9, 3, 9, 6, 3, 3, 6, 3, 12, 9, 9, 3, 9, 6, 12, 12, 9, 12, 3, 12, 12, 9, 12, 3, 3, 3, 6

Offset: 0

Carl R. White, Jan 16 2008

From Robert Israel, Jun 08 2018: (Start)
For n >= 6, a(n) is 3, 6, 9 or 12.
For k >= 2, a(5*k) = a(5*k+1) = a(5*k+3). (End)

			6! = 720 decimal = 330 quindecimal, so a(6) = 3.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000142, A136690, A136691, A136692, A136693, A136694, A136695, A136696, A008904, A136697, A136698, A136699, A136700, A136702.

a:= 1: b:= 0: R[0]:= 1:
for n from 1 to 100 do
   alpha:= padic:-ordp(n,3);
   beta:= padic:-ordp(n,5);
   a:= a * n/3^alpha/5^beta;
   b:= b + alpha - beta;
   R[n]:= a * 3 &^ b mod 15;
od:
seq(R[n],n=0..100); # Robert Israel, Jun 08 2018

nzd[x_]:=If[x[[-1,1]]==0,x[[-2,1]],x[[-1,1]]]; Table[nzd[Split[IntegerDigits[n!,15]]],{n,0,100}] (* Harvey P. Dale, Jul 11 2023 *)

cp's OEIS Frontend

A008904 a(n) is the final nonzero digit of n!.

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A136691 Final nonzero digit of n! in base 4.

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A136695 Final nonzero digit of n! in base 8.

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A136698 Final nonzero digit of n! in base 12.

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A136692 Final nonzero digit of n! in base 5.

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A136694 Final nonzero digit of n! in base 7.

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A136696 Final nonzero digit of n! in base 9.

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A136697 Final nonzero digit of n! in base 11.

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