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

0,11

Comments

Also the highest power of 10 dividing n! (different from A054899). - Hieronymus Fischer, Jun 18 2007

a(n) = (n - A053824(n))/4. - Lekraj Beedassy, Nov 01 2010

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

References

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

Links

T. D. Noe and 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.

S. Ikeda, K. Matsuoka, On transcendental numbers generated by certain integer sequences, Siauliai Math. Semin., 8 (16) 2013, 63-69.

S-C Liu, J. C.-C. Yeh, Catalan numbers modulo 2^k, J. Int. Seq. 13 (2010), 10.5.4, eq (5).

A. M. Oller-Marcén. A new look at the trailing zeros of n!, arXiv:0906.4868v1 [math.NT], 2009.

A. M. Oller-Marcen, 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

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Prog

(Haskell)
a027868 n = sum $ takeWhile (> 0) $ map (n `div`) $ tail a000351_list
-- Reinhard Zumkeller, Oct 31 2012
(PARI) 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
(PARI) a(n)=valuation(n!, 5) \\ Charles R Greathouse IV, Nov 08 2012
(PARI) apply( A027868(n)=(n-sumdigits(n, 5))\4, [0..99]) \\ M. F. Hasler, Dec 27 2019
(Python)
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
(Python)
def A027868(n): return 0 if n<5 else n//5 + A027868(n//5) # David Radcliffe, Jun 26 2016
(Magma) [Valuation(Factorial(n), 5): n in [0..80]]; // Bruno Berselli, Oct 11 2021

Crossrefs

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. A008904

Sequence in context: A154099 A105511 A187183 * A060384 A105564 A241766

Adjacent sequences: A027865 A027866 A027867 * A027869 A027870 A027871

Keyword

nonn,base,nice,easy

Author

Warut Roonguthai

Extensions

Examples added by Hieronymus Fischer, Jun 06 2012

Status

approved