%I A027868
%S A027868 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,
%T A027868 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,
%U A027868 13,14,14,14,14,14,15,15,15,15,15,16,16,16,16,16,18,18,18,18,18,19
%N A027868 Number of trailing zeros in n!; highest power of 5 dividing n!.
%C A027868 Also the highest power of 10 dividing n! (different from A054899). - 
               Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
%H A027868 T. D. Noe, <a href="b027868.txt">Table of n, a(n) for n=0..1000</a>
%H A027868 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Factorial.html">Link to a section of The World of Mathematics.</a>
%F A027868 Floor[n/5] + floor[n/25] + floor[n/125] + floor[n/625] + ....
%F A027868 Sum [ n/5^i ] from i=1 to infinity.
%F A027868 a(n)=(n-A053824(n))/4
%F A027868 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
%F A027868 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
%F A027868 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
%F A027868 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
%F A027868 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
%F A027868 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
%F A027868 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
%F A027868 a(n)<=(n-1)/4; equality holds for powers of 5. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Aug 13 2007
%F A027868 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
%F A027868 lim inf (n/4-a(n))=1/4, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Aug 13 2007
%F A027868 lim sup (n/4-log_5(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Aug 13 2007
%F A027868 lim sup (a(n+1)-a(n)-log_5(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Aug 13 2007
%F A027868 a(n) <= A027869(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jan 27 2008
%t A027868 Table[t = 0; p = 5; While[s = Floor[n/p]; t = t + s; s > 0, p *= 5]; 
               t, {n, 0, 100} ]
%Y A027868 See A000966 for the missing numbers. Cf. A011371 and A054861 for analogues 
               involving powers of 2 and 3.
%Y A027868 Cf. A054899, A007953, A112765, A067080, A098844, A132027.
%Y A027868 Cf. A067080, A098844, A132029, A054999.
%Y A027868 Sequence in context: A008648 A154099 A105511 this_sequence A060384 A105564 
               A025811
%Y A027868 Adjacent sequences: A027865 A027866 A027867 this_sequence A027869 A027870 
               A027871
%K A027868 nonn,base,nice,easy
%O A027868 0,11
%A A027868 Warut Roonguthai (warut822(AT)yahoo.com)