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Search: id:A054893
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| A054893 |
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Floor[n/4] + floor[n/16] + floor[n/64] + floor[n/256] + .... |
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+0
3
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| 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; listen)
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OFFSET
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0,9
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COMMENT
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Different from highest power of 4 dividing n! (see A090616).
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FORMULA
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a(n)=(n-A053737(n))/3
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). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
a(k*4^m)=k*(4^m-1)/3, for 0<=k<4, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
Asymptotic behavior: a(n)=n/3+O(log(n)), a(n+1)-a(n)=O(log(n)); this follows from the inequalities below. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
a(n)<=(n-1)/3; equality holds true for powers of 4. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
a(n)>=(n-3)/3-floor(log_4(n)); equality holds true for n=4^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
lim inf (n/3-a(n))=1/3, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
lim sup (n/3-log_4(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
lim sup (a(n+1)-a(n)-log_4(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
G.f.: g(x)=sum{k>0, x^(4^k)/(1-x^(4^k))}/(1-x). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
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MATHEMATICA
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Table[t = 0; p = 4; While[s = Floor[n/p]; t = t + s; s > 0, p *= 4]; t, {n, 0, 100} ]
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CROSSREFS
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Cf. A011371, A054861, A027868.
Cf. A011371, A054861, A027868, A054899, A067080, A098844, A132028.
Sequence in context: A088004 A070548 A132011 this_sequence A090617 A053693 A068063
Adjacent sequences: A054890 A054891 A054892 this_sequence A054894 A054895 A054896
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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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EXTENSIONS
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Edited by Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 15 2007
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