login
This site is supported by donations to The OEIS Foundation.
Logo

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A027868 Number of trailing zeros in n!; highest power of 5 dividing n!. 29
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 (list; graph; refs; listen; history; internal format)
OFFSET

0,11

COMMENTS

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

a(n) = (n - A053824(n))/4. [From Lekraj Beedassy (blekraj(AT)yahoo.com), 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. [From Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 03 2010]

REFERENCES

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. [From J.M. Grau Ribas (grau(AT)uniovi.es), Feb 14 2010]

A. M. Oller-MarcAeeeen. A new lookat the trailing zeroes of n!. arXiv:0906.4868v1 [math.NT]. [From J.M. Grau Ribas (grau(AT)uniovi.es), Feb 14 2010]

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

FORMULA

Floor[n/5] + floor[n/25] + floor[n/125] + floor[n/625] + ....

Sum [ n/5^i ] from i=1 to infinity.

a(n)=(n-A053824(n))/4

G.f.: g(x)=sum{k>0, x^(5^k)/(1-x^(5^k))}/(1-x). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007

a(n)=sum{5<=k<=n, sum{j|k,j>=5, floor(log_5(j))-floor(log_5(j-1))}}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007

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. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007

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))}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007

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). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

a(k*5^m)=k*(5^m-1)/4, for 0<=k<5, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

Asymtotic behavior: a(n)=n/4+O(log(n)), a(n+1)-a(n)=O(log(n)), which follows from the inequalities below. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

a(n)<=(n-1)/4; equality holds for powers of 5. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

a(n)>=n/4-1-floor(log_5(n)); equality holds for n=5^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

lim inf (n/4-a(n))=1/4, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

lim sup (n/4-log_5(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

lim sup (a(n+1)-a(n)-log_5(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

a(n) <= A027869(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 27 2008

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}] (* RGWv *)

CROSSREFS

See A000966 for the missing numbers. Cf. A011371 and A054861 for analogues involving powers of 2 and 3.

Cf. A054899, A007953, A112765, A067080, A098844, A132027.

Cf. A067080, A098844, A132029, A054999.

Sequence in context: A154099 A105511 A187183 * A060384 A105564 A025811

Adjacent sequences:  A027865 A027866 A027867 * A027869 A027870 A027871

KEYWORD

nonn,base,nice,easy

AUTHOR

Warut Roonguthai (warut822(AT)yahoo.com)

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Puzzles | Hot | Classics
Recent Additions | More pages | Superseeker | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified February 23 08:31 EST 2012. Contains 206628 sequences.