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a(n) = Sum_{j > 0} floor(n/4^j).
6

%I #28 Jan 08 2024 09:00:28

%S 0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,10,

%T 10,10,10,11,11,11,11,12,12,12,12,13,13,13,13,15,15,15,15,16,16,16,16,

%U 17,17,17,17,18,18,18,18,21,21,21,21,22,22,22,22,23,23,23,23,24,24,24,24

%N a(n) = Sum_{j > 0} floor(n/4^j).

%C Different from highest power of 4 dividing n! (see A090616).

%H Hieronymus Fischer, <a href="/A054893/b054893.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = floor(n/4) + floor(n/16) + floor(n/64) + floor(n/256) + ...

%F a(n) = (n - A053737(n))/3.

%F From _Hieronymus Fischer_, Sep 15 2007: (Start)

%F a(n) = a(floor(n/4)) + floor(n/4).

%F a(4*n) = a(n) + n.

%F a(n*4^m) = a(n) + n*(4^m-1)/3.

%F a(k*4^m) = k*(4^m-1)/3, for 0 <= k < 4, m >= 0.

%F Asymptotic behavior:

%F a(n) = n/3 + O(log(n)),

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

%F a(n) <= (n-1)/3; equality holds true for powers of 4.

%F a(n) >= (n-3)/3 - floor(log_4(n)); equality holds true for n = 4^m - 1, m>0. lim inf (n/3 - a(n)) = 1/3, for n-->oo.

%F lim sup (n/3 - log_4(n) - a(n)) = 0, for n-->oo.

%F lim sup (a(n+1) - a(n) - log_4(n)) = 0, for n-->oo.

%F G.f.: (1/(1-x))*Sum_{k > 0} x^(4^k)/(1-x^(4^k)). (End)

%F Partial sums of A235127. - _R. J. Mathar_, Jul 08 2021

%e a(10^0) = 0.

%e a(10^1) = 2.

%e a(10^2) = 32.

%e a(10^3) = 330.

%e a(10^4) = 3331.

%e a(10^5) = 33330.

%e a(10^6) = 333330.

%e a(10^7) = 3333329.

%e a(10^8) = 33333328.

%e a(10^9) = 333333326.

%t Table[t=0; p=4; While[s=Floor[n/p]; t=t+s; s>0, p *= 4]; t, {n,0,100}]

%t Table[Total[Floor/@(n/NestList[4#&,4,6])],{n,0,80}] (* _Harvey P. Dale_, Jun 12 2022 *)

%o (Magma)

%o function A054893(n)

%o if n eq 0 then return n;

%o else return A054893(Floor(n/4)) + Floor(n/4);

%o end if; return A054893;

%o end function;

%o [A054893(n): n in [0..103]]; // _G. C. Greubel_, Feb 09 2023

%o (SageMath)

%o def A054893(n):

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

%o else: return A054893(n//4) + (n//4)

%o [A054893(n) for n in range(104)] # _G. C. Greubel_, Feb 09 2023

%o (PARI) a(n) = (n - sumdigits(n,4))/3; \\ _Kevin Ryde_, Jan 08 2024

%Y Cf. A053737, A235127 (first differences).

%Y Cf. A011371, A027868, A054861, A054899, A067080, A090616, A098844, A132028.

%K nonn,easy

%O 0,9

%A _Henry Bottomley_, May 23 2000

%E Edited by _Hieronymus Fischer_, Sep 15 2007

%E Examples added by _Hieronymus Fischer_, Jun 06 2012