%I A007953
%S A007953 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,
%T A007953 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,
%U A007953 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
%N A007953 Digital sum (i.e. sum of digits) of n.
%C A007953 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.
%C A007953 For n<100 equal to (floor(n/10)+ n mod 10)=A076314(n). - Hieronymus Fischer 
               (Hieronymus.Fischer(AT)gmx.de), Jun 17 2007
%C A007953 a(n) = A138530(n,10) for n > 9. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 26 2008
%C A007953 a(A058369(n))=A004159(A058369(n)); a(A000290(n))=A004159(n). [From Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 25 2009]
%D A007953 K. Atanassov, On the 16-th Smarandache Problem, Notes on Number Theory 
               and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 1, 
               36-38.
%D A007953 K. Atanassov, On Some of Smarandache's Problems, American Research Press, 
               1999, 12-15.
%D A007953 F. Smarandache, "Only Problems, not Solutions!", Xiquan Publ., Phoenix-Chicago, 
               1993.
%H A007953 N. J. A. Sloane, <a href="b007953.txt">Table of n, a(n) for n = 0..10000</
               a>
%H A007953 K. Atanassov, <a href="http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf">
               On Some of Smarandache's Problems</a>
%H A007953 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/
               ">Smarandache Notions Journal</a>
%H A007953 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">
               Only Problems, Not Solutions!</a>.
%H A007953 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DigitSum.html">Digit Sum</a>
%F A007953 a(n) <= 9(log_10(n)+1). - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Mar 24 2006
%F A007953 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
%F A007953 G.f. g(x)=sum{k>0, (x^k-x^(k+10^k)-9x^(10^k))/(1-x^(10^k))}/(1-x). - 
               Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 17 2007
%F A007953 a(n)=n-9*sum{10<=k<=n, sum{j|k,j>=10, floor(log_10(j))-floor(log_10(j-1))}}. 
               - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 17 2007
%F A007953 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. 
               - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
%F A007953 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))}. 
               - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
%F A007953 a(n)=n-9*sum_{0<k<=floor(log_10(n))} a(floor(n/10^k))*10^(k-1). - Hieronymus 
               Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
%F A007953 a(n)<=9*(1+floor(log_10(n)), equality holds for n=10^m-1, m>0. - Hieronymus 
               Fischer (Hieronymus.Fischer(AT)gmx.de), Oct 06 2007
%F A007953 lim sup (a(n)-9*log_10(n))=0 for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Oct 06 2007
%F A007953 lim inf (a(n+1)-a(n)+9*log_10(n))=1 for n-->oo. - Hieronymus Fischer 
               (Hieronymus.Fischer(AT)gmx.de), Oct 06 2007
%e A007953 a(123)=1+2+3=6, a(9875)=9+8+7+5=29.
%p A007953 A007953 := proc(n) local t1,t2; t1 := n; t2 := 0; while t1 <> 0 do t2 
               := t2 + (t1 mod 10); t1 := floor(t1/10); od: t2; end;
%p A007953 for n from 0 to 8 do seq(n^1+j^1, j=0..9 ); od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 06 2006
%t A007953 Table[Sum[DigitCount[n][[i]]*i, {i, 1, 9}], {n, 1, 50}] - Stefan Steinerberger 
               (stefan.steinerberger(AT)gmail.com), Mar 24 2006
%t A007953 Table[Plus @@ IntegerDigits@n, {n, 0, 87}] (* or *)
%t A007953 Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 9}]] &, {0}, 2] 
               (* Robert G. Wilson v Jul 27 2006 *)
%o A007953 (PARI) a(n)=if(n<1,0,if(n%10,a(n-1)+1,a(n/10)))
%Y A007953 Cf. A003132, A055012, A055013, A055014, A055015, A010888, A007954, A031347.
%Y A007953 Cf. A055017, A076313, A076314, A007953, A003132, A054899.
%Y A007953 a(A051885(n)) = n.
%Y A007953 Cf. A138470, A138471, A138472.
%Y A007953 A000120. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 25 2009]
%Y A007953 Sequence in context: A131650 A033930 A076314 this_sequence A080463 A114570 
               A115026
%Y A007953 Adjacent sequences: A007950 A007951 A007952 this_sequence A007954 A007955 
               A007956
%K A007953 nonn,base,nice,easy
%O A007953 0,3
%A A007953 R. Muller
%E A007953 More terms from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 
               17 2007