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A054895 a(n) = Sum_{k>0} floor(n/6^k). 11

%I #33 Feb 09 2023 14:20:11

%S 0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,

%T 5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,

%U 11,11,11,12,12,12,12,12,12,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16

%N a(n) = Sum_{k>0} floor(n/6^k).

%C Different from the highest power of 6 dividing n! (cf. A054861). - _Hieronymus Fischer_, Aug 14 2007

%C Partial sums of A122841. - _Hieronymus Fischer_, Jun 06 2012

%H Hieronymus Fischer, <a href="/A054895/b054895.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = floor(n/6) + floor(n/36) + floor(n/216) + floor(n/1296) + ...

%F a(n) = (n - A053827(n))/5.

%F From _Hieronymus Fischer_, Aug 14 2007: (Start)

%F a(n) = a(floor(n/6)) + floor(n/6).

%F a(6*n) = n + a(n).

%F a(n*6^m) = n*(6^m-1)/5 + a(n).

%F a(k*6^m) = k*(6^m-1)/5, for 0 <= k < 6, m >= 0.

%F Asymptotic behavior:

%F a(n) = (n/5) + O(log(n)).

%F a(n+1) - a(n) = O(log(n)); this follows from the inequalities below.

%F a(n) <= (n-1)/5; equality holds for powers of 6.

%F a(n) >= ((n-5)/5) - floor(log_6(n)); equality holds for n=6^m-1, m>0.

%F lim inf (n/5 - a(n)) = 1/5, for n-->oo.

%F lim sup (n/5 - log_6(n) - a(n)) = 0, for n-->oo.

%F lim sup (a(n+1) - a(n) - log_6(n)) = 0, for n-->oo.

%F G.f.: (1/(1-x))*Sum_{k > 0} x^(6^k)/(1-x^(6^k)). (End)

%e a(10^0) = 0.

%e a(10^1) = 1.

%e a(10^2) = 18.

%e a(10^3) = 197.

%e a(10^4) = 1997.

%e a(10^5) = 19996.

%e a(10^6) = 199995.

%e a(10^7) = 1999995.

%e a(10^8) = 19999994.

%e a(10^9) = 199999993.

%t Table[t=0; p=6; While[s=Floor[n/p]; t=t+s; s>0, p *= 6]; t, {n,0,100}]

%o (Haskell)

%o a054895 n = a054895_list !! n

%o a054895_list = scanl (+) 0 a122841_list

%o -- _Reinhard Zumkeller_, Nov 10 2013

%o (Magma)

%o function A054895(n)

%o if n eq 0 then return n;

%o else return A054895(Floor(n/6)) + Floor(n/6);

%o end if; return A054895;

%o end function;

%o [A054895(n): n in [0..100]]; // _G. C. Greubel_, Feb 09 2023

%o (SageMath)

%o def A054895(n):

%o if (n==0): return 0

%o else: return A054895(n//6) + (n//6)

%o [A054895(n) for n in range(104)] # _G. C. Greubel_, Feb 09 2023

%Y Cf. A011371 and A054861 for analogs involving powers of 2 and 3.

%Y Cf. A053827, A054861, A054899, A067080, A098844, A122841, A132030.

%K nonn

%O 0,13

%A _Henry Bottomley_, May 23 2000

%E An incorrect formula was deleted by _N. J. A. Sloane_, Nov 18 2008

%E Examples added by _Hieronymus Fischer_, Jun 06 2012

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)