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A054893 a(n) = floor(n/4) + floor(n/16) + floor(n/64) + floor(n/256) + .... 5
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, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

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

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

FORMULA

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

From Hieronymus Fischer, Sep 15 2007: (Start)

Recurrence:

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

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

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

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

Asymptotic behavior:

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

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

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

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.

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

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

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

Partial sums of A235127. - R. J. Mathar, Jul 08 2021

EXAMPLE

a(100)=32.

a(10^3)=330.

a(10^4)=3331.

a(10^5)=33330.

a(10^6)=333330.

a(10^7)=3333329.

a(10^8)=33333328.

a(10^9)=333333326.

MATHEMATICA

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

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

CROSSREFS

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

Sequence in context: A070548 A209628 A132011 * A090617 A053693 A330324

Adjacent sequences: A054890 A054891 A054892 * A054894 A054895 A054896

KEYWORD

nonn,changed

AUTHOR

Henry Bottomley, May 23 2000

EXTENSIONS

Edited by Hieronymus Fischer, Sep 15 2007

Examples added by Hieronymus Fischer, Jun 06 2012

STATUS

approved

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Last modified November 30 02:32 EST 2022. Contains 358431 sequences. (Running on oeis4.)