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Search: id:A054861
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| A054861 |
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Highest power of 3 dividing n!. |
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+0
34
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| 0, 0, 0, 1, 1, 1, 2, 2, 2, 4, 4, 4, 5, 5, 5, 6, 6, 6, 8, 8, 8, 9, 9, 9, 10, 10, 10, 13, 13, 13, 14, 14, 14, 15, 15, 15, 17, 17, 17, 18, 18, 18, 19, 19, 19, 21, 21, 21, 22, 22, 22, 23, 23, 23, 26, 26, 26, 27, 27, 27, 28, 28, 28, 30, 30, 30, 31, 31, 31, 32, 32, 32, 34, 34, 34, 35, 35
(list; graph; listen)
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OFFSET
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0,7
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COMMENT
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Also the number of trailing zeros in the base-3 representation of n!. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
Also the highest power of 6 dividing n!. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
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FORMULA
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a(n) = floor[n/3] + floor[n/9] + floor[n/27] + floor[n/81] + ....
G.f.: g(x)=sum{k>0, x^(3^k)/(1-x^(3^k))}/(1-x). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
a(n)=sum{3<=k<=n, sum{j>=3,j|k, floor(log_3(j))-floor(log_3(j-1))}}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
G.f.: g(x)=L[b(k)](x)/(1-x) where L[b(k)](x)=sum{k>=0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k)=1, if k>1 is a power of 3, else b(k)=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
G.f.: g(x)=sum{k>0, c(k)*x^k}/(1-x), where c(k)=sum{j>1,j|k, floor(log_3(j))-floor(log_3(j-1))}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
Recurrence: a(n)=floor(n/3)+a(floor(n/3)); a(3*n)=n+a(n); a(n*3^m)=n*(3^m-1)/2+a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(k*3^m)=k*(3^m-1)/2, for 0<=k<3, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
Asymptotic behavior: a(n)=n/2+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)/2; equality holds for powers of 3. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(n)>=(n-2)/2-floor(log_3(n)); equality holds for n=3^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim inf (n/2-a(n))=1/2, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim sup (n/2-log_3(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_3(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
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MATHEMATICA
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Table[t = 0; p = 3; While[s = Floor[n/p]; t = t + s; s > 0, p *= 2]; t, {n, 0, 100} ]
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CROSSREFS
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a(n+1)=sum(k=1, n, A007949(k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 24 2002
a(n)=(n-A053735(n))/2
Cf. A011371 for analogue involving powers of 2. See also A027868.
See A004128 for a(3n).
Cf. A054895, A067080, A098844, A132027.
Sequence in context: A106160 A007614 A113402 this_sequence A086227 A079438 A123050
Adjacent sequences: A054858 A054859 A054860 this_sequence A054862 A054863 A054864
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KEYWORD
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easy,nonn
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), May 22 2000
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