login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A054899 a(n) = Sum_{k>0} floor(n/10^k). 59
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
Eric Weisstein's World of Mathematics, Multifactorial
FORMULA
a(n) = floor(n/10) + floor(n/100) + floor(n/1000) + ...
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_{k=10..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_{k=0..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_{k=0..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
(Magma)
m:=10;
function a(n) // a = A054899, m = 10
if n eq 0 then return 0;
else return a(Floor(n/m)) + Floor(n/m);
end if; end function;
[a(n): n in [0..103]]; // G. C. Greubel, Apr 28 2023
(SageMath)
m=10 # a = A054899
def a(n): return 0 if (n==0) else a(n//m) + (n//m)
[a(n) for n in range(104)] # G. C. Greubel, Apr 28 2023
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).
Sequence in context: A132272 A179051 A324160 * A061217 A102684 A337637
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)