cp's OEIS Frontend

A191610 Possible number of trailing zeros in k!.

%I A191610 #49 Feb 18 2025 11:38:46
%S A191610 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,
%T A191610 32,33,34,35,37,38,39,40,41,43,44,45,46,47,49,50,51,52,53,55,56,57,58,
%U A191610 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
%N A191610 Possible number of trailing zeros in k!.
%C A191610 Equivalently, possible values of 10-adic valuation of k!. - _Joerg Arndt_, Sep 21 2020
%H A191610 Reinhard Zumkeller, <a href="/A191610/b191610.txt">Table of n, a(n) for n = 1..10000</a>
%F A191610 a(n) ~ 5*n/4. - _Vaclav Kotesovec_, Sep 21 2020
%F A191610 G.f.: 1/(1-x) * Sum_{k>=0} x^(5^k)/(1-x^5^k). - _Joerg Arndt_, Sep 21 2020
%F A191610 a(n) = Sum_{k>=0} floor((n-1)/5^k). - _Clark Kimberling_, Feb 17 2025
%e A191610 3 is a term because 15! = 1307674368000 has 3 trailing 0's.
%e A191610 5 is not a term because 24! has 4 trailing 0's, but 25! has 6 trailing 0's.
%t A191610 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 *)
%t A191610 Table[Sum[Floor[(n - 1)/5^k], {k, 0, Floor[Log[5, n]]}], {n, 1, 200}] (* _Clark Kimberling_, Feb 17 2025 *)
%o A191610 (Haskell)
%o A191610 a191610 1 = 0
%o A191610 a191610 n = sum $ takeWhile (> 0) $ map ((n - 1) `div`) a000351_list
%o A191610 -- _Reinhard Zumkeller_, Oct 31 2012
%o A191610 (Python)
%o A191610 # requires Python 3.2 and higher
%o A191610 from itertools import accumulate
%o A191610 from sympy import multiplicity
%o A191610 A191610 = [0]+list(accumulate(multiplicity(5,n) for n in range(5,10**3,5)))
%o A191610 # _Chai Wah Wu_, Sep 05 2014
%Y A191610 Cf. A027868, A000351, A055457 (first differences).
%Y A191610 Complement of A000966.
%Y A191610 Cf. A380662, A381239.
%K A191610 nonn
%O A191610 1,3
%A A191610 _José María Grau Ribas_, Jun 09 2011