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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: A059995 A132272 A179051 * 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 June 25 20:26 EDT 2017. Contains 288730 sequences.