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Search: id:A054896
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| A054896 |
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Sum_{k>0} floor(n/7^k). |
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11
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| 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13
(list; graph; listen)
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OFFSET
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0,15
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COMMENT
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Highest power of 7 dividing n!.
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FORMULA
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floor[n/7] + floor[n/49] + floor[n/343] + floor[n/2401] + ....
a(n)=(n-A053828(n))/6
a(n)= -1 + Sum_{k=0..n} 1/147*{-20*[n mod 7]+[(n+1) mod 7]+[(n+2) mod 7]+[(n+3) mod 7]+[(n+4) mod 7]+[(n+5) mod 7]+22*[(n+6) mod 7]},with n>=0. - Paolo P. Lava (ppl(AT)spl.at), May 15 2007
Recurrence: a(n)=floor(n/7)+a(floor(n/7)); a(7*n)=n+a(n); a(n*7^m)=n*(7^m-1)/6+a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(k*7^m)=k*(7^m-1)/6, for 0<=k<7, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
Asymptotic behavior: a(n)=n/6+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)/6; equality holds for powers of 7. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(n)>=(n-6)/6-floor(log_7(n)); equality holds for n=7^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim inf (n/6-a(n))=1/6, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim sup (n/6-log_7(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_7(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
G.f.: g(x)=sum{k>0, x^(7^k)/(1-x^(7^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 = 7; While[s = Floor[n/p]; t = t + s; s > 0, p *= 7]; 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. A054895, A054899, A067080, A098844, A132031.
Sequence in context: A115338 A133877 A132270 this_sequence A052364 A052374 A003074
Adjacent sequences: A054893 A054894 A054895 this_sequence A054897 A054898 A054899
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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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