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%I #24 Feb 09 2023 14:18:19
%S 0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,
%T 4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,
%U 10,10,10,10,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13
%N a(n) = Sum_{k>0} floor(n/7^k).
%C Highest power of 7 dividing n!.
%H Hieronymus Fischer, <a href="/A054896/b054896.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = floor(n/7) + floor(n/49) + floor(n/343) + floor(n/2401) + ...
%F a(n) = (n - A053828(n))/6.
%F From _Hieronymus Fischer_, Aug 14 2007: (Start)
%F a(n) = a(floor(n/7)) + floor(n/7).
%F a(7*n) = n + a(n).
%F a(n*7^m) = a(n) + n*(7^m-1)/6.
%F a(k*7^m) = k*(7^m-1)/6, for 0 <= k < 7, m >= 0.
%F Asymptotic behavior:
%F a(n) = n/6 + O(log(n)).
%F a(n+1) - a(n) = O(log(n)); this follows from the inequalities below.
%F a(n) <= (n-1)/6; equality holds for powers of 7.
%F a(n) >= (n-6)/6 - floor(log_7(n)); equality holds for n=7^m-1, m>0. -
%F lim inf (n/6 - a(n)) = 1/6, for n-->oo.
%F lim sup (n/6 - log_7(n) - a(n)) = 0, for n-->oo.
%F lim sup (a(n+1) - a(n) - log_7(n)) = 0, for n-->oo.
%F G.f.: (1/(1-x))*Sum_{k > 0} x^(7^k)/(1-x^(7^k)). (End)
%F Partial sums of A214411. - _R. J. Mathar_, Jul 08 2021
%e a(10^0) = 0.
%e a(10^1) = 1.
%e a(10^3) = 16.
%e a(10^3) = 164.
%e a(10^4) = 1665.
%e a(10^5) = 16662.
%e a(10^6) = 166664.
%e a(10^7) = 1666661.
%e a(10^8) = 16666662.
%e a(10^9) = 166666661
%t Table[t=0; p=7; While[s=Floor[n/p]; t=t+s; s>0, p *= 7]; t, {n,0,100}]
%o (Magma)
%o function A054896(n)
%o if n eq 0 then return n;
%o else return A054896(Floor(n/7)) + Floor(n/7);
%o end if; return A054896;
%o end function;
%o [A054896(n): n in [0..100]]; // _G. C. Greubel_, Feb 09 2023
%o (SageMath)
%o def A054896(n):
%o if (n==0): return 0
%o else: return A054896(n//7) + (n//7)
%o [A054896(n) for n in range(101)] # _G. C. Greubel_, Feb 09 2023
%Y Cf. A011371 and A054861 for analogs involving powers of 2 and 3.
%Y Cf. A053828, A054893, A054895, A054899, A067080, A098844, A132031, A214411.
%K nonn
%O 0,15
%A _Henry Bottomley_, May 23 2000
%E Examples added by _Hieronymus Fischer_, Jun 06 2012
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