A008904 -id:A008904 - cp's OEIS frontend

A027868 Number of trailing zeros in n!; highest power of 5 dividing n!.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 18, 18, 18, 18, 18, 19

Offset: 0

Warut Roonguthai

Also the highest power of 10 dividing n! (different from A054899). - Hieronymus Fischer, Jun 18 2007
Alternatively, a(n) equals the expansion of the base-5 representation A007091(n) of n (i.e., where successive positions from right to left stand for 5^n or A000351(n)) under a scale of notation whose successive positions from right to left stand for (5^n - 1)/4 or A003463(n); for instance, n = 7392 has base-5 expression 2*5^5 + 1*5^4 + 4*5^3 + 0*5^2 + 3*5^1 + 2*5^0, so that a(7392) = 2*781 + 1*156 + 4*31 + 0*6 + 3*1 + 2*0 = 1845. - Lekraj Beedassy, Nov 03 2010
Partial sums of A112765. - Hieronymus Fischer, Jun 06 2012
Also the number of trailing zeros in A000165(n) = (2*n)!!. - Stefano Spezia, Aug 18 2024

			a(100)  = 24.
a(10^3) = 249.
a(10^4) = 2499.
a(10^5) = 24999.
a(10^6) = 249998.
a(10^7) = 2499999.
a(10^8) = 24999999.
a(10^9) = 249999998.
a(10^n) = 10^n/4 - 3 for 10 <= n <= 15 except for a(10^14) = 10^14/4 - 2. - _M. F. Hasler_, Dec 27 2019

M. Gardner, "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, 1978, pp. 50-65.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
David S. Hart, James E. Marengo, Darren A. Narayan, and David S. Ross, On the number of trailing zeros in n!, College Math. J., 39(2):139-145, 2008.
Enrique Pérez Herrero, Trailing Zeros in n!, Psychedelic Geometry Blogspot.
Soichi Ikeda and Kaneaki Matsuoka, On transcendental numbers generated by certain integer sequences, Siauliai Math. Semin., 8 (16) 2013, 63-69.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Shu-Chung Liu and Jean C.-C. Yeh, Catalan numbers modulo 2^k, J. Int. Seq. 13 (2010), 10.5.4, eq (5).
Antonio M. Oller-Marcén, A new look at the trailing zeros of n!, arXiv:0906.4868v1 [math.NT], 2009.
Antonio M. Oller-Marcén and J. Maria Grau, On the Base-b Expansion of the Number of Trailing Zeros of b^k!, J. Int. Seq. 14 (2011) 11.6.8
Eric Weisstein's World of Mathematics, Factorial.
Index entries for sequences related to factorial numbers

See A000966 for the missing numbers. See A011371 and A054861 for analogs involving powers of 2 and 3.
Cf. A054899, A007953, A112765, A067080, A098844, A132027, A067080, A098844, A132029, A054999, A112765, A191610, A000351.
Cf. also A000142, A004154.
Cf. A000165, A008904.

a027868 n = sum $ takeWhile (> 0) $ map (n `div`) $ tail a000351_list
-- Reinhard Zumkeller, Oct 31 2012

[Valuation(Factorial(n), 5): n in [0..80]]; // Bruno Berselli, Oct 11 2021

0, seq(add(floor(n/5^i),i=1..floor(log[5](n))), n=1..100); # Robert Israel, Nov 13 2014

Table[t = 0; p = 5; While[s = Floor[n/p]; t = t + s; s > 0, p *= 5]; t, {n, 0, 100} ]
Table[ IntegerExponent[n!], {n, 0, 80}] (* Robert G. Wilson v *)
zOF[n_Integer?Positive]:=Module[{maxpow=0},While[5^maxpow<=n,maxpow++];Plus@@Table[Quotient[n,5^i],{i,maxpow-1}]]; Attributes[zOF]={Listable}; Join[{0},zOF[ Range[100]]] (* Harvey P. Dale, Apr 11 2022 *)

a(n)={my(s);while(n\=5,s+=n);s} \\ Charles R Greathouse IV, Nov 08 2012, edited by M. F. Hasler, Dec 27 2019

a(n)=valuation(n!,5) \\ Charles R Greathouse IV, Nov 08 2012

apply( A027868(n)=(n-sumdigits(n,5))\4, [0..99]) \\ M. F. Hasler, Dec 27 2019

from sympy import multiplicity
A027868, p5 = [0,0,0,0,0], 0
for n in range(5,10**3,5):
    p5 += multiplicity(5,n)
    A027868.extend([p5]*5) # Chai Wah Wu, Sep 05 2014

def A027868(n): return 0 if n<5 else n//5 + A027868(n//5) # David Radcliffe, Jun 26 2016

from sympy.ntheory.factor_ import digits
def A027868(n): return n-sum(digits(n,5)[1:])>>2 # Chai Wah Wu, Oct 18 2024

a(n) = Sum_{i>=1} floor(n/5^i).
a(n) = (n - A053824(n))/4.
From Hieronymus Fischer, Jun 25 2007 and Aug 13 2007, edited by M. F. Hasler, Dec 27 2019: (Start)
G.f.: g(x) = Sum_{k>0} x^(5^k)/(1-x^(5^k))/(1-x).
a(n) = Sum_{k=5..n} Sum_{j|k, j>=5} (floor(log_5(j)) - floor(log_5(j-1))).
G.f.: g(x) = L[b(k)](x)/(1-x) where L[b(k)](x) = Sum_{k>=0} b(k)*x^k/(1-x^k) is a Lambert series with b(k) = 1, if k>1 is a power of 5, else b(k) = 0.
G.f.: g(x) = Sum_{k>0} c(k)*x^k/(1-x), where c(k) = Sum_{j>1, j|k} floor(log_5(j)) - floor(log_5(j - 1)).
Recurrence:
a(n) = floor(n/5) + a(floor(n/5));
a(5*n) = n + a(n);
a(n*5^m) = n*(5^m-1)/4 + a(n).
a(k*5^m) = k*(5^m-1)/4, for 0 <= k < 5, m >= 0.
Asymptotic behavior:
a(n) = n/4 + O(log(n)),
a(n+1) - a(n) = O(log(n)), which follows from the inequalities below.
a(n) <= (n-1)/4; equality holds for powers of 5.
a(n) >= n/4 - 1 - floor(log_5(n)); equality holds for n = 5^m-1, m > 0.
lim inf (n/4 - a(n)) = 1/4, for n -> oo.
lim sup (n/4 - log_5(n) - a(n)) = 0, for n -> oo.
lim sup (a(n+1) - a(n) - log_5(n)) = 0, for n -> oo.
(End)
a(n) <= A027869(n). - Reinhard Zumkeller, Jan 27 2008
10^a(n) = A000142(n) / A004154(n). - Reinhard Zumkeller, Nov 24 2012
a(n) = Sum_{k=1..floor(n/2)} floor(log_5(n/k)). - Ammar Khatab, Feb 01 2025

Examples added by Hieronymus Fischer, Jun 06 2012

A004154 a(n) = n! with trailing zeros omitted.

Original entry on oeis.org

1, 1, 2, 6, 24, 12, 72, 504, 4032, 36288, 36288, 399168, 4790016, 62270208, 871782912, 1307674368, 20922789888, 355687428096, 6402373705728, 121645100408832, 243290200817664, 5109094217170944, 112400072777760768, 2585201673888497664, 62044840173323943936

Offset: 0

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 0..250
Index entries for sequences related to factorial numbers

Cf. A000142, A004151, A008904 (mod 10).

a004154 = a004151 . a000142
a004154_list = scanl (\u v -> a004151 $ u * v) 1 [1..]
-- Reinhard Zumkeller, Nov 24 2012

[Factorial(n)/10^Valuation(Factorial(n), 5): n in [0..30]]; // Vincenzo Librandi, Oct 16 2014

a:= n-> (f-> f/10^padic[ordp](f,10))(n!):
seq(a(n), n=0..29);  # Alois P. Heinz, Dec 29 2021

Array[#!//.x_/;x~Mod~5==0:>x/10&,22]  (* Giorgos Kalogeropoulos, Aug 17 2020 *)
Join[{1,1,2,6,24},Table[FromDigits[Flatten[Most[Split[IntegerDigits[n!]]]]],{n,5,30}]] (* or *) Table[n!/10^IntegerExponent[n!,10],{n,0,30}] (* Harvey P. Dale, Feb 13 2024 *)

a(n)=n!/10^valuation(n!,5) \\ M. F. Hasler, Oct 16 2014

from sympy import factorial
from sympy.ntheory.factor_ import digits
def A004154(n): return factorial(n)//10**(n-sum(digits(n,5)[1:])>>2) # Chai Wah Wu, Oct 18 2024

from itertools import count, islice
def agen(): # generator of terms
    f = 1
    for n in count(1):
        yield f
        while n%10 == 0: n //= 10
        f *= n
        while f%10 == 0: f //= 10
print(list(islice(agen(), 25))) # Michael S. Branicky, Apr 11 2025

a(n) = A000142(n) / 10^A027868(n). - Reinhard Zumkeller, Nov 24 2012
a(n+1) = A004151((n+1)*a(n)). - Reinhard Zumkeller, Nov 24 2012, corrected by M. F. Hasler, Oct 16 2014
a(n) = A004151(A000142(n)) = A000142(n)/A011557(A112765(n)), or A122840 instead of A112765. - M. F. Hasler, Oct 16 2014

A136690 Final nonzero digit of n! in base 3.

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2

Offset: 0

Carl R. White, Jan 16 2008

			6! = 720 decimal = 222200 ternary, so a(6) = 2.

Cf. A000142, A127110.

Other bases: A136691, A136692, A136693, A136694, A136695, A136696, A008904, A136697, A136698, A136699, A136700, A136701, A136702, A212307.

f[n_] := Mod[6 Times @@ (Rest[ FoldList[{1 + #1[[1]], #2! 2^(#1[[1]] #2)} &, {0, 0}, Reverse[ IntegerDigits[n, 3]]]]), 10][[2]]; # /. {0 -> 1} & /@ Mod[Table[f@n, {n, 0, 104}], 3] (* Robert G. Wilson v, Apr 17 2010 *)
fnzd[n_]:=Module[{sidn3=Split[IntegerDigits[n!,3]]},If[MemberQ[ Last[ sidn3],0], sidn3[[-2,1]], sidn3[[-1,1]]]]; Array[fnzd,110,0] (* Harvey P. Dale, May 03 2018 *)

a(n) = vecsum([bittest(220,b) |b<-digits(n,9)])%2 + 1; \\ Kevin Ryde, Dec 03 2022

From David Radcliffe, Sep 03 2021: (Start)
a(n) = (n! / A060828(n)) mod 3;
a(n) = 1 + (A189672(n) mod 2);
a(6*n) = a(6*n+1) = a(2*n);
a(6*n+2) = 3 - a(2*n);
a(6*n+3) = a(6*n+4) = 3 - a(2*n+1);
a(6*n+5) = a(2*n+1).
(End)
a(n) = A008904(A127110(n)). - Michel Marcus, Sep 04 2021
From Kevin Ryde, Dec 03 2022: (Start)
a(n) = 1 if n written in base 9 has an even number of digits {2,3,4,6,7}; and otherwise a(n) = 2.
Fixed point of the morphism 1 -> 1,1,2,2,2,1,2,2,1; 2 -> 2,2,1,1,1,2,1,1,2; starting from 1.
(End)
a(n) = A212307(n) mod 3. - Ridouane Oudra, Sep 25 2024

More terms from Robert G. Wilson v, Apr 17 2010

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

cp's OEIS Frontend

A027868 Number of trailing zeros in n!; highest power of 5 dividing n!.

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A004154 a(n) = n! with trailing zeros omitted.

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A136690 Final nonzero digit of n! in base 3.

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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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