A053830 Sum of digits of (n written in base 9).

0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9

Offset: 0

Author

Henry Bottomley, Mar 28 2000

Keywords

• base
• nonn

Comments

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8}, 1->{1,2,3,4,5,6,7,8,9}, 2->{2,3,4,5,6,7,8,9,10}, etc. - Robert G. Wilson v, Jul 27 2006

Examples

			a(20) = 2+2 = 4 because 20 is written as 22 base 9.
From _Omar E. Pol_, Feb 23 2010: (Start)
It appears that this can be written as a triangle (see the conjecture in the entry A000120):
0;
1,2,3,4,5,6,7,8;
1,2,3,4,5,6,7,8,9,2,3,4,5,6,7,8,9,10,3,4,5,6,7,8,9,10,11,4,5,6,7,8,9,10,11,...
where the rows converge to A173529. (End)
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..59049
• Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
• Robert Walker, Self Similar Sloth Canon Number Sequences.
• Eric Weisstein's World of Mathematics, Digit Sum.

Crossrefs

Cf. A000120, A007953, A053735, A053737, A053824, A053828, A231684, A231685, A231686, A231687.

Cf. A173529. - Omar E. Pol, Feb 23 2010

Programs

Magma

[&+Intseq(n, 9):n in [0..100]]; // Marius A. Burtea, Aug 24 2019

Mathematica

Table[Plus @@ IntegerDigits[n, 9], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 8}]] &, {0}, 3] (* Robert G. Wilson v, Jul 27 2006 *)

PARI

a(n)=if(n<1,0,if(n%9,a(n-1)+1,a(n/9)))

Formula

From Benoit Cloitre, Dec 19 2002: (Start)

a(0) = 0, a(9n+i) = a(n) + i for 0 <= i <= 8;

a(n) = n - 8*Sum_{k>=1} floor(n/9^k) = n - 8*A054898(n). (End)

a(n) = A138530(n,9) for n > 8. - Reinhard Zumkeller, Mar 26 2008

a(n) = Sum_{k>=0} A031087(n,k). - Philippe Deléham, Oct 21 2011

a(0) = 0; a(n) = a(n - 9^floor(log_9(n))) + 1. - Ilya Gutkovskiy, Aug 24 2019

Sum_{n>=1} a(n)/(n*(n+1)) = 9*log(9)/8 (Shallit, 1984). - Amiram Eldar, Jun 03 2021