A045550 -id:A045550 - 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

A045547 Numbers whose factorial has '2' as its final nonzero digit.

Original entry on oeis.org

2, 5, 6, 8, 14, 19, 34, 35, 36, 38, 40, 41, 43, 47, 50, 51, 53, 62, 67, 74, 84, 85, 86, 88, 90, 91, 93, 97, 109, 110, 111, 113, 115, 116, 118, 122, 129, 132, 145, 146, 148, 150, 151, 153, 162, 167, 174, 177, 180, 181, 183, 189, 194, 200, 201, 203, 212, 217

Offset: 1

Jeff Burch

From Robert Israel, Dec 16 2016: (Start)
If k is in the sequence, then:
if k == 0 (mod 5), k+1 is in the sequence;
if k == 1 (mod 5), k+1 is in A045548;
if k == 2 (mod 5), k+1 is in A045549;
if k == 3 (mod 5), k+1 is in A045550. (End)

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to final digits of numbers

Cf. A008904, A045548, A045549, A045550.

count:= 0:
r:= 1:
for n from 2 while count < 100 do
  r:= r*n;
  if r mod 10 = 0 then r:= r/10^padic:-ordp(r, 5) fi;
  if r mod 10 = 2 then count:= count+1; A[count]:= n fi;
od: seq(A[i], i=1..100); # Robert Israel, Dec 16 2016

f[ n_Integer, m_Integer ] := (c = 0; p = 1; While[ d = Floor[ n/5^p ]; d > 0, c = c + d; p++ ]; Mod[ n!/10^c, m ] ); Select[ Range[ 250 ], f[ #, 10 ] == 2 & ]
Join[{2},Select[Range[5,220],Most[Split[IntegerDigits[#!]]][[-1,1]] == 2&]] (* Harvey P. Dale, May 04 2016 *)
f[n_] := Mod[6 Times @@ (Rest[ FoldList[{1 + #1[[1]], #2! 2^(#1[[1]] #2)} &, {0, 0}, Reverse[ IntegerDigits[n, 5]]]]), 10][[2]] (* after Jacob A. Siehler & Greg Dresden in A008904 *); f[0] = f[1] = 1; Select[ Range[150], f[#] == 2 &] (* Robert G. Wilson v, Dec 28 2016 *)

lnz(n)=if(n<2, return(1)); my(m=Mod(1,5)); for(k=2,n, m*=k/10^valuation(k,5)); lift(chinese(Mod(0,2),m))
is(n)=lnz(n)==2 \\ Charles R Greathouse IV, Dec 16 2016

list(lim)=my(v=List(),m=Mod(1,5)); for(k=2,lim, m*=k/10^valuation(k,5); if(m==2, listput(v, k))); Vec(v) \\ Charles R Greathouse IV, Dec 16 2016

from functools import reduce
from itertools import count, islice
from sympy.ntheory.factor_ import digits
def A045547_gen(startvalue=1): # generator of terms
    return filter(lambda n:2==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(n,5)[-1:0:-1])))*6%10, count(max(startvalue,1)))
A045547_list = list(islice(A045547_gen(),30)) # Chai Wah Wu, Dec 07 2023

A045548 Numbers whose factorial has '4' as its final nonzero digit.

Original entry on oeis.org

4, 7, 20, 21, 23, 25, 26, 28, 37, 42, 49, 52, 55, 56, 58, 64, 69, 75, 76, 78, 87, 92, 99, 100, 101, 103, 112, 117, 124, 134, 135, 136, 138, 140, 141, 143, 147, 152, 155, 156, 158, 164, 169, 179, 182, 195, 196, 198, 202, 205, 206, 208, 214, 219

Offset: 1

Jeff Burch

From Robert Israel, Dec 16 2016: (Start)
If n is in the sequence, then:
if n == 0 (mod 5), n+1 is in the sequence;
if n == 1 (mod 5), n+1 is in A045550;
if n == 2 (mod 5), n+1 is in A045547;
if n == 3 (mod 5), n+1 is in A045549. (End)

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to final digits of numbers

Cf. A000142, A008904, A045547, A045549, A045550.

count:= 0:
r:= 1:
for n from 2 while count < 100 do
  r:= r*n;
  if r mod 10 = 0 then r:= r/10^padic:-ordp(r, 5) fi;
  if r mod 10 = 4 then count:= count+1; A[count]:= n fi;
od: seq(A[i], i=1..100); # Robert Israel, Dec 16 2016

f[n_] := Mod[6 Times @@ (Rest[ FoldList[{1 + #1[[1]], #2! 2^(#1[[1]] #2)} &, {0, 0}, Reverse[ IntegerDigits[n, 5]]]]), 10][[2]] (* after Jacob A. Siehler & Greg Dresden in A008904 *); f[0] = f[1] = 1; Select[ Range[150], f[#] == 4 &] (* Robert G. Wilson v, Dec 28 2016 *)

from itertools import count, islice
from functools import reduce
from sympy.ntheory.factor_ import digits
def A045548_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:4==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), count(max(startvalue,1)))
A045548_list = list(islice(A045548_gen(),30)) # Chai Wah Wu, Dec 07 2023

A045549 Numbers whose factorial has '6' as its final nonzero digit.

Original entry on oeis.org

3, 12, 17, 24, 29, 32, 45, 46, 48, 59, 60, 61, 63, 65, 66, 68, 72, 79, 82, 95, 96, 98, 104, 107, 120, 121, 123, 127, 130, 131, 133, 139, 144, 159, 160, 161, 163, 165, 166, 168, 172, 175, 176, 178, 187, 192, 199, 209, 210, 211, 213, 215, 216, 218

Offset: 1

Jeff Burch

From Robert Israel, Dec 16 2016: (Start)
If n is in the sequence, then:
if n == 0 (mod 5), n+1 is in the sequence;
if n == 1 (mod 5), n+1 is in A045547;
if n == 2 (mod 5), n+1 is in A045550;
if n == 3 (mod 5), n+1 is in A045548. (End)

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to final digits of numbers

Cf. A008904, A045547, A045548, A045550.

count:= 0:
r:= 1:
for n from 2 while count < 100 do
  r:= r*n;
  if r mod 10 = 0 then r:= r/10^padic:-ordp(r,5) fi;
  if r mod 10 = 6 then count:= count+1; A[count]:= n fi;
od:
seq(A[i],i=1..100); # Robert Israel, Dec 16 2016

Join[{3}, Select[Range[5,250], Most[Split[IntegerDigits[#!]]][[-1, 1]] == 6 &]] (* Vincenzo Librandi, Dec 16 2016 *)
f[n_] := Mod[6 Times @@ (Rest[ FoldList[{1 + #1[[1]], #2! 2^(#1[[1]] #2)} &, {0, 0}, Reverse[ IntegerDigits[n, 5]]]]), 10][[2]] (* after Jacob A. Siehler & Greg Dresden in A008904 *); f[0] = f[1] = 1; Select[ Range[150], f[#] == 6 &] (* Robert G. Wilson v, Dec 28 2016 *)
Select[Range[250],With[{f=#!},Drop[IntegerDigits[f],-IntegerExponent[f]][[-1]]]==6&] (* Harvey P. Dale, Sep 27 2024 *)

from itertools import count, islice
from functools import reduce
from sympy.ntheory.factor_ import digits
def A045549_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:6==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), count(max(startvalue,2)))
A045549_list = list(islice(A045549_gen(),30)) # Chai Wah Wu, Dec 07 2023

A058376 Where the race of the count of final nonzero digit of k! changes, starting at k=2.

Original entry on oeis.org

2, 16, 50, 80, 88, 108, 110, 264, 273, 291, 326, 336, 669, 671, 678, 685, 718, 721, 738, 764, 773, 791, 826, 836, 1433, 1435, 1558, 1560, 1616, 1629, 1636, 1694, 1696, 1764, 1773, 1791, 1826, 1836, 1928, 1935, 1968, 1971, 1988, 2014, 2023, 2041, 2076, 2086

Offset: 1

Robert G. Wilson v, Dec 19 2000

			a(1) = 2 to start the race. At 15! the number of final twos is 5 and so is the number of eights. But at 16, eights now lead twos, so a(2) = 16 to reflect this fact.
When k=10000 is reached, the count stands at 2509 twos, 2486 fours, 2494 sixes, and 2510 eights.

Cf. A045547, A045548, A045549, A045550.

f[ n_Integer, m_Integer ] := (c = 0; p = 1; While[ d = Floor[ n/5^p ]; d > 0, c = c + d; p++ ]; Mod[ n!/10^c, m ]); a = Table[ 0, {4} ]; r = 4; Do[ b = f[ n, 10 ]; Switch[ b, 2, a[ [ 1 ] ]++, 4, a[ [ 2 ] ]++, 6, a[ [ 3 ] ]++, 8, a[ [ 4 ] ]++ ]; If[ a[ [ b/2 ] ] > a[ [ r/2 ] ], r = b; Print[ n ] ], {n, 2, 10^4} ]

from functools import reduce
from itertools import count, islice
from sympy.ntheory.factor_ import digits
def A058376_gen(): # generator of terms
    a, k, i = [0]*4, 0, 1
    for n in count(2):
        m = (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(n,5)[-1:0:-1])))*6%10>>1)-1
        a[m] += 1
        if a[m] > k:
            if m!=i:
                yield n
            i, k = m, a[m]
A058376_list = list(islice(A058376_gen(),48)) # Chai Wah Wu, Dec 07 2023

Offset 1 from Michel Marcus, Jul 25 2021

A374015 Residue modulo 5 of n! divided by the highest power of 10 which divides n!.

Original entry on oeis.org

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

Offset: 0

Cezary Glowacz, Jun 25 2024

The sequence is not eventually periodic. This because by induction on k the eventual period must be a multiple of 5^k for every k.
a(5^k) = 2^k mod 5.
From Cezary Glowacz, Feb 07 2025: (Start)
The proportions p(m,s) of counts of pairs of consecutive terms s among a(1),...,a(m) converge to equidistribution (and as an immediate consequence, so do proportions of individual terms).
This can be seen, for example, by stating p(5^(4(n+1)+1)-1,s) as affine functions of p(5^(4n+1)-1,t) and examining the convergence of p(5^(4n+1)-1,u) to the equidistribution. Then, p(m,s) converges to the equidistribution because the maximum over s of the absolute values of deviations from 1/16 of p(m,s) for m>k*5^(4n+1)-1 is less than the corresponding maximum over t for p(5^(4n+1)-1,t) plus 2/(5^(4n+1)) + 1/k.
Consecutive terms 1,2,3 do not occur, so that triples do not have a similar equidistribution.
(End)
If n > 0 is not divisible by 5, a(n) == n * a(n-1) (mod 5). - Robert Israel, Jul 05 2024

			a(5) = 1*2*3*4*5/10 mod 5 = 2.

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

Cf. A004154, A010874, A145679.

Cf. A045549, A045547, A045550, A045548.

a:= n-> (f-> irem(f/10^padic[ordp](f, 10), 5))(n!):
seq(a(n), n=0..105);  # Alois P. Heinz, Jun 25 2024

a[n_]:=Mod[n!/10^IntegerExponent[n!, 10],5]; Array[a,106,0] (* Stefano Spezia, Jun 25 2024 *)

a(n)=if(n>4, my(k=n\5); return(lift((n%5)!*a(k)*Mod(2,5)^k))); n!%5 \\ Charles R Greathouse IV, Jan 24 2025

v=[[((1,1,2,1,4)[j]*2**(i*j))%5 for j in range(5)] for i in range(4)]
def a(n):
    c,p=0,1
    while n: c,n,p=(c+1)%4,n//5,(v[c][n%5]*p)%5
    return(p) # Cezary Glowacz, Feb 05 2025

a(n) = A010874(A004154(n)).

cp's OEIS Frontend

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

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A045547 Numbers whose factorial has '2' as its final nonzero digit.

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A045548 Numbers whose factorial has '4' as its final nonzero digit.

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A045549 Numbers whose factorial has '6' as its final nonzero digit.

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A058376 Where the race of the count of final nonzero digit of k! changes, starting at k=2.

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A374015 Residue modulo 5 of n! divided by the highest power of 10 which divides n!.

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