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Search: id:A054895
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| A054895 |
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Sum_{k>0} floor(n/6^k). |
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+0
10
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| 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16
(list; graph; listen)
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OFFSET
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0,13
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COMMENT
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Different from the highest power of 6 dividing n! (cf. A054861). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
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FORMULA
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floor[n/6] + floor[n/36] + floor[n/216] + floor[n/1296] + ....
a(n)=(n-A053827(n))/5
a(n)= -1 + Sum_{k=0..n} 1/90*{-14*[k mod 6]+[(k+1) mod 6]+[(k+2) mod 6]+[(k+3) mod 6]+[(k+4) mod 6]+16*[(k+5) mod 6]}, with n>=0. - Paolo P. Lava (ppl(AT)spl.at), May 15 2007
Recurrence: a(n)=floor(n/6)+a(floor(n/6)); a(6*n)=n+a(n); a(n*6^m)=n*(6^m-1)/5+a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(k*6^m)=k*(6^m-1)/5, for 0<=k<6, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
Asymptotic behavior: a(n)=n/5+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)/5; equality holds for powers of 6. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(n)>=(n-5)/5-floor(log_6(n)); equality holds for n=6^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim inf (n/5-a(n))=1/5, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim sup (n/5-log_6(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_6(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
G.f.: g(x)=sum{k>0, x^(6^k)/(1-x^(6^k))}/(1-x). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
Also, floor((n^6-1)/(5*n^5)) will produce this sequence. Moreover, floor[(n^6-n^5)/(6n^5-5n^4)] will produce this sequence as well. - Mohammad K. Azarian (azarian(AT)evansville.edu), Nov 08 2007
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MATHEMATICA
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Table[t = 0; p = 6; While[s = Floor[n/p]; t = t + s; s > 0, p *= 6]; 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. A054861, A054899, A067080, A098844, A132030.
Sequence in context: A138194 A133876 A097992 this_sequence A137588 A033271 A004052
Adjacent sequences: A054892 A054893 A054894 this_sequence A054896 A054897 A054898
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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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