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%I #109 Nov 03 2023 06:42:44
%S 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,
%T 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,
%U 14,14,14,14,14,15,15,15,15,15,16,16,16,16,16,18,18,18,18,18,19
%N Number of trailing zeros in n!; highest power of 5 dividing n!.
%C Also the highest power of 10 dividing n! (different from A054899). - _Hieronymus Fischer_, Jun 18 2007
%C a(n) = (n - A053824(n))/4. - _Lekraj Beedassy_, Nov 01 2010
%C 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
%C Partial sums of A112765. - _Hieronymus Fischer_, Jun 06 2012
%D 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.
%H Hieronymus Fischer, <a href="/A027868/b027868.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from T. D. Noe)
%H David S. Hart, James E. Marengo, Darren A. Narayan and David S. Ross, <a href="http://www.jstor.org/stable/27646601">On the number of trailing zeros in n!</a>, College Math. J., 39(2):139-145, 2008.
%H Enrique Pérez Herrero, <a href="http://psychedelic-geometry.blogspot.com/2009/08/trailing-zeros-in-n.html">Trailing Zeros in n!</a>, Psychedelic Geometry Blogspot.
%H S. Ikeda, K. Matsuoka, <a href="http://siauliaims.su.lt/pdfai/2013/Iked-Mats-2013.pdf">On transcendental numbers generated by certain integer sequences</a>, Siauliai Math. Semin., 8 (16) 2013, 63-69.
%H S-C Liu, J. C.-C. Yeh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Liu2/liu6.html">Catalan numbers modulo 2^k</a>, J. Int. Seq. 13 (2010), 10.5.4, eq (5).
%H A. M. Oller-Marcén. <a href="http://arxiv.org/abs/0906.4868">A new look at the trailing zeros of n!</a>, arXiv:0906.4868v1 [math.NT], 2009.
%H A. M. Oller-Marcen, J. Maria Grau, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Oller/oller3.html">On the Base-b Expansion of the Number of Trailing Zeros of b^k!</a>, J. Int. Seq. 14 (2011) 11.6.8
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Factorial.html">Factorial</a>.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F a(n) = Sum_{i>=1} floor(n/5^i).
%F a(n) = (n - A053824(n))/4.
%F From _Hieronymus Fischer_, Jun 25 2007 and Aug 13 2007, edited by _M. F. Hasler_, Dec 27 2019: (Start)
%F G.f.: g(x) = Sum_{k>0} x^(5^k)/(1-x^(5^k))/(1-x).
%F a(n) = Sum_{k=5..n} Sum_{j|k, j>=5} (floor(log_5(j)) - floor(log_5(j-1))).
%F G.f.: g(x) = L[b(k)](x)/(1-x)
%F 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.
%F G.f.: g(x) = Sum_{k>0} c(k)*x^k/(1-x),
%F where c(k) = Sum_{j>1, j|k} floor(log_5(j)) - floor(log_5(j - 1)).
%F Recurrence:
%F a(n) = floor(n/5) + a(floor(n/5));
%F a(5*n) = n + a(n);
%F a(n*5^m) = n*(5^m-1)/4 + a(n).
%F a(k*5^m) = k*(5^m-1)/4, for 0 <= k < 5, m >= 0.
%F Asymptotic behavior:
%F a(n) = n/4 + O(log(n)),
%F a(n+1) - a(n) = O(log(n)), which follows from the inequalities below.
%F a(n) <= (n-1)/4; equality holds for powers of 5.
%F a(n) >= n/4 - 1 - floor(log_5(n)); equality holds for n = 5^m-1, m > 0.
%F lim inf (n/4 - a(n)) = 1/4, for n -> oo.
%F lim sup (n/4 - log_5(n) - a(n)) = 0, for n -> oo.
%F lim sup (a(n+1) - a(n) - log_5(n)) = 0, for n -> oo.
%F (End)
%F a(n) <= A027869(n). - _Reinhard Zumkeller_, Jan 27 2008
%F 10^a(n) = A000142(n) / A004154(n). - _Reinhard Zumkeller_, Nov 24 2012
%e a(100) = 24.
%e a(10^3) = 249.
%e a(10^4) = 2499.
%e a(10^5) = 24999.
%e a(10^6) = 249998.
%e a(10^7) = 2499999.
%e a(10^8) = 24999999.
%e a(10^9) = 249999998.
%e 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
%p 0, seq(add(floor(n/5^i),i=1..floor(log[5](n))), n=1..100); # _Robert Israel_, Nov 13 2014
%t Table[t = 0; p = 5; While[s = Floor[n/p]; t = t + s; s > 0, p *= 5]; t, {n, 0, 100} ]
%t Table[ IntegerExponent[n!], {n, 0, 80}] (* _Robert G. Wilson v_ *)
%t 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 *)
%o (Haskell)
%o a027868 n = sum $ takeWhile (> 0) $ map (n `div`) $ tail a000351_list
%o -- _Reinhard Zumkeller_, Oct 31 2012
%o (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
%o (PARI) a(n)=valuation(n!,5) \\ _Charles R Greathouse IV_, Nov 08 2012
%o (PARI) apply( A027868(n)=(n-sumdigits(n,5))\4, [0..99]) \\ _M. F. Hasler_, Dec 27 2019
%o (Python)
%o from sympy import multiplicity
%o A027868, p5 = [0,0,0,0,0], 0
%o for n in range(5,10**3,5):
%o p5 += multiplicity(5,n)
%o A027868.extend([p5]*5) # _Chai Wah Wu_, Sep 05 2014
%o (Python)
%o def A027868(n): return 0 if n<5 else n//5 + A027868(n//5) # _David Radcliffe_, Jun 26 2016
%o (Magma) [Valuation(Factorial(n), 5): n in [0..80]]; // _Bruno Berselli_, Oct 11 2021
%Y See A000966 for the missing numbers. See A011371 and A054861 for analogs involving powers of 2 and 3.
%Y Cf. A054899, A007953, A112765, A067080, A098844, A132027, A067080, A098844, A132029, A054999, A112765, A191610, A000351.
%Y Cf. also A000142, A004154.
%Y Cf. A008904
%K nonn,base,nice,easy
%O 0,11
%A _Warut Roonguthai_
%E Examples added by _Hieronymus Fischer_, Jun 06 2012
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