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 A054899 a(n) = Sum_{k>0} floor(n/10^k). 58
 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; refs; listen; history; text; internal format)
 OFFSET 0,21 COMMENTS 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 06 2012 LINKS Hieronymus Fischer, Table of n, a(n) for n = 0..10000 Eric Weisstein's World of Mathematics, Multifactorial. FORMULA floor[n/10] + floor[n/100] + floor[n/1000] + floor[n/10000] + .... a(n) = (n - A007953(n))/9. From Hieronymus Fischer, Jun 14 2007, 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) EXAMPLE 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. MATHEMATICA Table[t = 0; p = 10; While[s = Floor[n/p]; t = t + s; s > 0, p *= 10]; t, {n, 0, 100} ] PROG (PARI) a(n)=my(s); while(n\=10, s+=n); s \\ Charles R Greathouse IV, Jul 19 2011 CROSSREFS Cf. A011371 and A054861 for analogs 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: A132272 A179051 A324160 * A061217 A102684 A156821 Adjacent sequences:  A054896 A054897 A054898 * A054900 A054901 A054902 KEYWORD nonn AUTHOR Henry Bottomley, May 23 2000 EXTENSIONS An incorrect g.f. was deleted by N. J. A. Sloane, Sep 13 2009 Examples added by Hieronymus Fischer, Jun 06 2012 STATUS approved

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Last modified November 22 13:47 EST 2019. Contains 329393 sequences. (Running on oeis4.)