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Search: id:A054899
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| A054899 |
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Sum {k>0, floor(n/10^k)}. |
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33
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| 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11
(list; graph; listen)
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OFFSET
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0,21
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COMMENT
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The old definition of this sequence was "Highest power of 10 dividing n!", but that is wrong (see A027868). For example, the highest power of 10 dividing 5!=120 is 1; however, a(5)=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
Highest power of 10 dividing the quotient of multifactorials product{k>=1, M(10^k,10^k*floor(n/10^k))}/product{k>=1, M(10^(k-1),10^(k-1)*floor(n/10^k))} where M(r,s) is the r-multifactorial (r>=1) of s which is defined by M(r,s)=s*M(r,s-r) for s>0, M(r,s)=1 for s<=0. - This is because that quotient of multifactorials evaluates to the product 10^floor(n/10)*10^floor(n/100)*... - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 14 2007
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LINKS
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Eric Weisstein's World of Mathematics, Multifactorial.
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FORMULA
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floor[n/10] + floor[n/100] + floor[n/1000] + floor[n/10000] + ....
a(n)=(n-A007953(n))/9
a(n) = sum {k>0, floor(n/10^k)}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 14 2007
a(n)=sum{10<=k<=n, sum{j|k,j>=10, floor(log_10(j))-floor(log_10(j-1))}}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
G.f.: g(x)=sum{k>0, x^(10^k)/(1-x^(10^k))}/(1-x). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
G.f. expressed in terms of Lambert series: 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 10, 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_10(j))-floor(log_10(j-1))}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
a(n)=sum_{0<=k<=floor(log_10(n))} ds_10(floor(n/10^k))*10^k - n where ds_10(x) = digital sum of x in base = 10. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
a(n)=sum_{0<=k<=floor(log_10(n))} A007953(floor(n/10^k))*10^k - n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
Recurrence: a(n)=floor(n/10)+a(floor(n/10)); a(10*n)=n+a(n); a(n*10^m)=n*(10^m-1)/9+a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
a(k*10^m)=k*(10^m-1)/9, for 0<=k<10, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
Asymtotic behavior: a(n)=n/9+O(log(n)), a(n+1)-a(n)=O(log(n)), which follows from the inequalities below. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
a(n)<=(n-1)/9; equality holds for powers of 10. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
a(n)>=n/9-1-floor(log_10(n)); equality holds for n=10^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
lim inf (n/9-a(n))=1/9, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
lim sup (n/9-log_10(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
lim sup (a(n+1)-a(n)-log_10(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
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MATHEMATICA
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Table[t = 0; p = 10; While[s = Floor[n/p]; t = t + s; s > 0, p *= 10]; 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.
Different from the highest power of 10 dividing n! (see A027868 for reference).
Cf. A027868.
Cf. A067080, A098844, A132027.
Sequence in context: A133880 A059995 A132272 this_sequence A061217 A102684 A025856
Adjacent sequences: A054896 A054897 A054898 this_sequence A054900 A054901 A054902
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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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