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A054893 Floor[n/4] + floor[n/16] + floor[n/64] + floor[n/256] + .... 5

%I

%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 Floor[n/4] + floor[n/16] + floor[n/64] + floor[n/256] + ....

%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)=(n-A053737(n))/3

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

%F Recurrence:

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

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

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

%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.: g(x) = sum{k>0, x^(4^k)/(1-x^(4^k))}/(1-x). (End)

%e a(100)=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} ]

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

%K nonn

%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

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Last modified September 29 21:59 EDT 2020. Contains 337432 sequences. (Running on oeis4.)