A000966 -id:A000966 - 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

A008905 Leading digit of n!.

Original entry on oeis.org

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

Offset: 0

Russ Cox

Kunoff proved that the distribution of terms of this sequence follows Benford's law, i.e., the asymptotic density of terms with value d (between 1 and 9) is log_10(1+1/d). - Amiram Eldar, Sep 23 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

Cf. A000966, A000142, A018799, A202021 (leading digit of (10^n)!), A213201.

a008905 = a000030 . a000142  -- Reinhard Zumkeller, Apr 08 2012

f[n_] := Quotient[n!, 10^Floor@ Log[10, n!]]; Array[f, 105, 0]

a(n) = A000030(A000142(n)). - Reinhard Zumkeller, Apr 08 2012

Two less-efficient Mathematica codings removed by Robert G. Wilson v, Nov 05 2010

A136774 n! never ends in this many 0's in base 16.

Original entry on oeis.org

62, 94, 110, 118, 126, 158, 174, 182, 190, 206, 214, 222, 230, 238, 254, 286, 302, 310, 318, 334, 342, 350, 358, 366, 382, 398, 406, 414, 422, 430, 446, 454, 462, 478, 510, 542, 558, 566, 574, 590, 598, 606, 614, 622, 638, 654, 662, 670, 678, 686, 702, 710

Offset: 1

Carl R. White, Jan 21 2008

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A000142, A136766, A136702, A000966, A055938, A096346, A136767, A136768, A136769, A136770, A136771, A136772, A136773.

is(n)=my(t=4*n+2,s=1-hammingweight(n)); while(s<0, s+=valuation(t+=2,2)); s>3 \\ Charles R Greathouse IV, Sep 22 2016

Conjecture: a(n) ~ 16n. This holds with probability 1 in a random model. - Charles R Greathouse IV, Sep 22 2016

A136767 n! never ends in this many 0's in base 4.

Original entry on oeis.org

6, 10, 14, 18, 22, 29, 30, 34, 38, 45, 46, 53, 57, 61, 62, 66, 70, 77, 78, 85, 89, 93, 94, 101, 105, 109, 113, 117, 124, 125, 126, 130, 134, 141, 142, 149, 153, 157, 158, 165, 169, 173, 177, 181, 188, 189, 190, 197, 201, 205, 209, 213, 220, 221, 225, 229, 236

Offset: 1

Carl R. White, Jan 21 2008

Cf. A000142, A136755, A136691.

Cf. A000966, A055938, A096346, A136768, A136769, A136770, A136771, A136772, A136773, A136774.

A136773 n! never ends in this many 0's in base 13.

Original entry on oeis.org

13, 27, 41, 55, 69, 83, 97, 111, 125, 139, 153, 167, 181, 182, 196, 210, 224, 238, 252, 266, 280, 294, 308, 322, 336, 350, 364, 365, 379, 393, 407, 421, 435, 449, 463, 477, 491, 505, 519, 533, 547, 548, 562, 576, 590, 604, 618, 632, 646, 660, 674, 688, 702

Offset: 1

Carl R. White, Jan 21 2008

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000142, A136763, A136699, A000966, A055938, A096346, A136767, A136768, A136769, A136770, A136771, A136772, A136774.

valp(n, p)=my(s); while(n\=p, s+=n); s
is(n)=my(t=(12*n-1)\13*13+13,s=valp(t,13)-n); while(s<0, s+=valuation(t+=13, 13)); s>0 \\ Charles R Greathouse IV, Oct 19 2016

A191610 Possible number of trailing zeros in k!.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 124, 125, 126, 127, 128, 130, 131, 132, 133, 134, 136

Offset: 1

José María Grau Ribas, Jun 09 2011

nonn

Equivalently, possible values of 10-adic valuation of k!. - Joerg Arndt, Sep 21 2020

			3 is a term because 15! = 1307674368000 has 3 trailing 0's.
5 is not a term because 24! has 4 trailing 0's, but 25! has 6 trailing 0's.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A027868, A000351, A055457 (first differences).
Complement of A000966.
Cf. A380662, A381239.

a191610 1 = 0
a191610 n = sum $ takeWhile (> 0) $ map ((n - 1) `div`) a000351_list
-- Reinhard Zumkeller, Oct 31 2012

zOF[n_Integer?Positive]:=Module[{maxpow=0},While[5^maxpow<=n,maxpow++];Plus@@Table[ Quotient[n,5^i],{i,maxpow-1}]]; Attributes[zOF]={Listable}; zOF[Range[1000]]//Union (* Harvey P. Dale, Dec 06 2023 *)
Table[Sum[Floor[(n - 1)/5^k], {k, 0, Floor[Log[5, n]]}], {n, 1, 200}] (* Clark Kimberling, Feb 17 2025 *)

# requires Python 3.2 and higher
from itertools import accumulate
from sympy import multiplicity
A191610 = [0]+list(accumulate(multiplicity(5,n) for n in range(5,10**3,5)))
# Chai Wah Wu, Sep 05 2014

a(n) ~ 5*n/4. - Vaclav Kotesovec, Sep 21 2020
G.f.: 1/(1-x) * Sum_{k>=0} x^(5^k)/(1-x^5^k). - Joerg Arndt, Sep 21 2020
a(n) = Sum_{k>=0} floor((n-1)/5^k). - Clark Kimberling, Feb 17 2025

cp's OEIS Frontend

A080066 First differences of A000966 (number of zeros that n! will never end in).

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

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A008905 Leading digit of n!.

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A136774 n! never ends in this many 0's in base 16.

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A136767 n! never ends in this many 0's in base 4.

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A136773 n! never ends in this many 0's in base 13.

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A191610 Possible number of trailing zeros in k!.

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A136768 n! never ends in this many 0's in base 7.