Search: id:A004128
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%I A004128
%S A004128 0,1,2,4,5,6,8,9,10,13,14,15,17,18,19,21,22,23,26,27,28,30,31,32,34,35,
%T A004128 36,40,41,42,44,45,46,48,49,50,53,54,55,57,58,59,61,62,63,66,67,68,70,
%U A004128 71,72,74,75,76,80,81,82,84,85,86,88,89,90,93,94,95,97,98,99,101,102
%N A004128 Sum_{k=1..n} floor(3n/3^k).
%C A004128 3-adic valuation of (3n)! - cf. A054861.
%C A004128 Denominators of expansion of (1-x)^{-1/3} are 3^a(n). Numerators are 
               in |A067622|.
%D A004128 Gary W. Adamson, in "Beyond Measure, A Guided Tour Through Nature, Myth 
               and Number", by Jay Kappraff, World Scientific, 2002, p. 356.
%H A004128 T. D. Noe, <a href="b004128.txt">Table of n, a(n) for n=0..1000</a>
%F A004128 a(n) = n+[n/3]+[n/9]+[n/27]+... = n+a([n/3]) = n+A054861(n) = A054861(3n) 
               = (3n-A053735(n))/2. - Henry Bottomley (se16(AT)btinternet.com), 
               May 01 2001
%F A004128 a(n)=sum{k>=0, floor(n/3^k)}. a(n)=sum{0<=k<=floor(log_3(n)), floor(n/
               3^k)}, n>=1. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Aug 14 2007
%F A004128 Recurrence: a(n)=n+a(floor(n/3)); a(3n)=3n+a(n); a(n*3^m)=3*n*(3^m-1)/
               2+a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 
               14 2007
%F A004128 a(k*3^m)=k*(3^(m+1)-1)/2, 0<=k<3, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Aug 14 2007
%F A004128 Asymptotic behavior: a(n)=3/2*n+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
%F A004128 a(n)<=(3n-1)/2; equality holds for powers of 3. - Hieronymus Fischer 
               (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
%F A004128 a(n)>=(3n-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
%F A004128 lim inf (3n/2-a(n))=1/2, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Aug 14 2007
%F A004128 lim sup (3n/2-log_3(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Aug 14 2007
%F A004128 lim sup (a(n+1)-a(n)-log_3(n))=1, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Aug 14 2007
%F A004128 G.f.: g(x)=sum{k>=0, x^(3^k)/(1-x^(3^k))}/(1-x). - Hieronymus Fischer 
               (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
%o A004128 (PARI) a(n)=local(s,t=1);while(t<=n,s+=n\t;t*=3);s - Michael Somos Feb 
               26 2004
%Y A004128 Cf. A004117, A001511, A051064, A055457.
%Y A004128 A051064(n) = a(n+1) - a(n). - Alford Arnold (Alford1940(AT)aol.com), 
               Jul 19 2000
%Y A004128 Cf. A054861, A067080, A098844, A132027, A005187, A054899.
%Y A004128 Sequence in context: A091529 A095775 A035063 this_sequence A023717 A043687 
               A087118
%Y A004128 Adjacent sequences: A004125 A004126 A004127 this_sequence A004129 A004130 
               A004131
%K A004128 nonn
%O A004128 0,3
%A A004128 N. J. A. Sloane (njas(AT)research.att.com).
%E A004128 Current definition suggested by Jason Earls (zevi_35711(AT)yahoo.com), 
               Jul 04 2001

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