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Search: id:A054897
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| A054897 |
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Sum_{k>0} floor(n/8^k). |
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4
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| 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; listen)
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OFFSET
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0,17
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COMMENT
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Different from the highest power of 8 dividing n!.
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FORMULA
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floor[n/8] + floor[n/64] + floor[n/512] + floor[n/4096] + ....
a(n)=(n-A053829(n))/7
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). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(k*8^m)=k*(8^m-1)/7, for 0<=k<8, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
Asymptotic behavior: a(n)=n/7+O(log(n)), a(n+1)-a(n)=O(log(n)); this follows from the inequalities below. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(n)<=(n-1)/7; equality holds for powers of 8. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(n)>=(n-7)/7-floor(log_8(n)); equality holds for n=8^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim inf (n/7-a(n))=1/7, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim sup (n/7-log_8(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim sup (a(n+1)-a(n)-log_8(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
G.f.: g(x)=sum{k>0, x^(8^k)/(1-x^(8^k))}/(1-x). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
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MATHEMATICA
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Table[t = 0; p = 8; While[s = Floor[n/p]; t = t + s; s > 0, p *= 8]; t, {n, 0, 100} ]
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CROSSREFS
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Cf. A011371 and A054861 for analogues involving powers of 2 and 3.
Cf. A054899, A067080, A098844, A132032.
Sequence in context: A132292 A110656 A104407 this_sequence A003108 A111898 A072746
Adjacent sequences: A054894 A054895 A054896 this_sequence A054898 A054899 A054900
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KEYWORD
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nonn
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), May 23 2000
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