0,3

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]

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.

F. Smarandache, "Only Problems, not Solutions!", Xiquan Publ., Phoenix-Chicago, 1993.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

K. Atanassov, On Some of Smarandache's Problems

M. L. Perez et al., eds., Smarandache Notions Journal

F. Smarandache, Only Problems, Not Solutions!.

Eric Weisstein's World of Mathematics, Digit Sum

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

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

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

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

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

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

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

lim sup (a(n)-9*log_10(n))=0 for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Oct 06 2007

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

a(123)=1+2+3=6, a(9875)=9+8+7+5=29.

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;

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

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 *)

(PARI) a(n)=if(n<1, 0, if(n%10, a(n-1)+1, a(n/10)))

Cf. A003132, A055012, A055013, A055014, A055015, A010888, A007954, A031347.

Cf. A055017, A076313, A076314, A007953, A003132, A054899.

a(A051885(n)) = n.

Cf. A138470, A138471, A138472.

A000120. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 25 2009]

Sequence in context: A131650 A033930 A076314 this_sequence A080463 A114570 A115026

Adjacent sequences: A007950 A007951 A007952 this_sequence A007954 A007955 A007956

nonn,base,nice,easy

R. Muller

More terms from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 17 2007