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A007953
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Digital sum (i.e. sum of digits) of n.
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409
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14, 15
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8,9}, 1->{1,2,3,4,5,6,7,8,9,10}, 2->{2,3,4,5,6,7,8,9,10,11}, etc. - Robert G. Wilson v Jul 27 2006.
For n<100 equal to (floor(n/10)+ n mod 10)=A076314(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 17 2007
a(n) = A138530(n,10) for n > 9. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 26 2008
a(A058369(n))=A004159(A058369(n)); a(A000290(n))=A004159(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 25 2009]
a(n) mod 2 = A179081(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 28 2010]
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REFERENCES
| K. Atanassov, On the 16-th Smarandache Problem, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 1, 36-38.
K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 12-15.
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178; http://www.combinatorics.org/Volume_18/PDF/v18i1p178.pdf.
J.-C. Puchta, J. Spilker, Altes und Neues zur Quersumme, Math. Semesterber, 49 (2002), 209-226.
V. Shevelev, Compact integers and factorials, Acta Arith. 126 (2007), no.3,195-236 (cf. pp.205-206).
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..10000
K. Atanassov, On Some of Smarandache's Problems
Eric Weisstein's World of Mathematics, Digit Sum
Wikipedia, Digit sum
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FORMULA
| a(n) <= 9(log_10(n)+1). - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 24 2006
a(0)=0, a(10n+i)=a(n)+i 0<=i<=9; a(n)=n-9*(sum(k>0, floor(n/10^k))=n-9*A054899(n). - Benoit Cloitre, Dec 19, 2002
From Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 17 2007: (Start)
G.f. g(x)=sum{k>0, (x^k-x^(k+10^k)-9x^(10^k))/(1-x^(10^k))}/(1-x).
a(n)=n-9*sum{10<=k<=n, sum{j|k,j>=10, floor(log_10(j))-floor(log_10(j-1))}}. (End)
From Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007: (Start)
The g.f. can be expressed in terms of a Lambert series, in that g(x)=(x/(1-x)-9*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, (1-9*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)=n-9*sum_{0<k<=floor(log_10(n))} a(floor(n/10^k))*10^(k-1). (End)
From Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Oct 06 2007: (Start)
a(n)<=9*(1+floor(log_10(n)), equality holds for n=10^m-1, m>0.
lim sup (a(n)-9*log_10(n))=0 for n-->oo.
lim inf (a(n+1)-a(n)+9*log_10(n))=1 for n-->oo. (End)
a(A051885(n)) = n.
a(n)<=9*log_10(n+1). -Vladimir Shevelev(shevelev(AT)bgu.ac.il), Jun 01 2011.
a(n) = a(n-1)+a(n-10)-a(n-11), for n < 100. - Alexander R. Povolotsky, Oct 09 2011.
a(n)=Sum_k>=0 {A031298(n,k)}. - From DELEHAM Philippe, Oct 21 2011.
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EXAMPLE
| a(123)=1+2+3=6, a(9875)=9+8+7+5=29.
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MAPLE
| A007953 := proc(n) add(d, d=convert(n, base, 10)) ; end proc: # R. J. Mathar, Mar 17 2011
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MATHEMATICA
| Table[Sum[DigitCount[n][[i]]*i, {i, 1, 9}], {n, 1, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 24 2006
Table[Plus @@ IntegerDigits@n, {n, 0, 87}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 9}]] &, {0}, 2] (* Robert G. Wilson v Jul 27 2006 *)
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PROG
| (PARI) a(n)=if(n<1, 0, if(n%10, a(n-1)+1, a(n/10)))
(PARI) A007953(n, b=10)={ my(s=(n=divrem(n, b))[2]); while(n[1]>=b, s+=(n=divrem(n[1], b))[2]); s+n[1]}
(Haskell)
a007953 n | n < 10 = n
| otherwise = a007953 n' + r where (n', r) = divMod n 10
-- Reinhard Zumkeller, Nov 04 2011, Mar 19 2011
(MAGMA) [ &+Intseq(n): n in [0..87] ]; // Bruno Berselli, May 26 2011
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CROSSREFS
| Cf. A003132, A055012, A055013, A055014, A055015, A010888, A007954, A031347, A055017, A076313, A076314, A007953, A003132, A054899, A138470, A138471, A138472.
Cf. A000120, A004426, A004427, A054683, A054684, A179082, A179083, A179084, A179085, A108971, A179987, A179988, A180018, A180019. [From Reinhard Zumkeller]
Sequence in context: A131650 A033930 A076314 * A080463 A114570 A115026
Adjacent sequences: A007950 A007951 A007952 * A007954 A007955 A007956
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KEYWORD
| nonn,base,nice,easy
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AUTHOR
| R. Muller
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EXTENSIONS
| More terms from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 17 2007
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