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A054897 a(n) = Sum_{k>0} floor(n/8^k). 4
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,17

COMMENTS

Different from the highest power of 8 dividing n!.

LINKS

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

FORMULA

floor[n/8] + floor[n/64] + floor[n/512] + floor[n/4096] + ....

a(n) = (n-A053829(n))/7.

From Hieronymus Fischer, Aug 14 2007: (Start)

Recurrence:

a(n) = floor(n/8) + a(floor(n/8));

a(8*n) = n + a(n);

a(n*8^m) = n*(8^m-1)/7 + a(n).

a(k*8^m) = k*(8^m-1)/7, for 0<=k<8, m>=0.

Asymptotic behavior:

a(n) = n/7 + O(log(n)),

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

a(n) <= (n-1)/7; equality holds for powers of 8.

a(n) >= (n-7)/7 - floor(log_8(n)); equality holds for n=8^m-1, m>0.

lim inf (n/7 - a(n)) = 1/7, for n-->oo.

lim sup (n/7 - log_8(n) - a(n)) = 0, for n-->oo.

lim sup (a(n+1) - a(n) - log_8(n)) = 0, for n-->oo.

G.f.: g(x) = sum{k>0, x^(8^k)/(1-x^(8^k))}/(1-x). (End)

EXAMPLE

a(100)=13.

a(10^3)=141.

a(10^4)=1427.

a(10^5)=14284.

a(10^6)=142855.

a(10^7)=1428569.

a(10^8)=14285710.

a(10^9)=142857138.

MATHEMATICA

Table[t = 0; p = 8; While[s = Floor[n/p]; t = t + s; s > 0, p *= 8]; t, {n, 0, 100} ]

CROSSREFS

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

Cf. A054899, A067080, A098844, A132032.

Sequence in context: A132292 A110656 A104407 * A261226 A003108 A279223

Adjacent sequences:  A054894 A054895 A054896 * A054898 A054899 A054900

KEYWORD

nonn

AUTHOR

Henry Bottomley, May 23 2000

EXTENSIONS

Examples added by Hieronymus Fischer, Jun 06 2012

STATUS

approved

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Last modified August 11 15:12 EDT 2020. Contains 336428 sequences. (Running on oeis4.)