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A054899
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Sum {k>0, floor(n/10^k)}.
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59
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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
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OFFSET
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0,21
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COMMENTS
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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, 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, Jun 14 2007
Partial sums of A122840. - Hieronymus Fischer, Jun 6 2012
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LINKS
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Hieronymus Fischer, Table of n, a(n) for n = 0..10000
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.
Contribution from Hieronymus Fischer, Jun 14 2007 and Jun 25 2007 and Aug 13 2007 (Start):
a(n) = sum {k>0, floor(n/10^k)}.
a(n) = sum{10<=k<=n, sum{j|k,j>=10, floor(log_10(j))-floor(log_10(j-1))}}.
G.f.: g(x) = sum{k>0, x^(10^k)/(1-x^(10^k))}/(1-x).
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.
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))}.
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.
a(n) = sum_{0<=k<=floor(log_10(n))} A007953(floor(n/10^k))*10^k - n.
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).
a(k*10^m) = k*(10^m-1)/9, for 0<=k<10, m>=0.
Asymptotic behavior:
a(n) = n/9 + O(log(n)),
a(n+1) - a(n) = O(log(n)), which follows from the inequalities below.
a(n) <= (n - 1)/9; equality holds for powers of 10.
a(n) >= n/9 - 1 - floor(log_10(n)); equality holds for n=10^m-1, m>0.
lim inf (n/9 - a(n)) = 1/9, for n-->oo.
lim sup (n/9 - log_10(n) - a(n)) = 0, for n-->oo.
lim sup (a(n+1) - a(n) - log_10(n)) = 0, for n-->oo. (End)
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EXAMPLE
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a(11)=1
a(111)=12.
a(1111)=123.
a(11111)=1234.
a(111111)=12345.
a(1111111)=123456.
a(11111111)=1234567.
a(111111111)=12345678.
a(1111111111)=123456789.
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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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PROG
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(PARI) a(n)=my(s); while(n\=10, s+=n); s \\ Charles R Greathouse IV, Jul 19 2011
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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, A067080, A098844, A132027, A122840.
Sequence in context: A059995 A132272 A179051 * A061217 A102684 A156821
Adjacent sequences: A054896 A054897 A054898 * A054900 A054901 A054902
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KEYWORD
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nonn,changed
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AUTHOR
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Henry Bottomley, May 23 2000
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EXTENSIONS
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An incorrect g.f. was deleted by N. J. A. Sloane, Sep 13 2009
Examples added by Hieronymus Fischer, Jun 06 2012
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STATUS
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approved
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