A053832 Sum of digits of n written in base 12.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 7, 8

Offset: 0

Author

Henry Bottomley, Mar 28 2000

Keywords

• base
• nonn

Comments

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8,9,10,11}, 1->{1,2,3,4,5,6,7,8,9,10,11,12}, 2->{2,3,4,5,6,7,8,9,10,11,12,13}, etc. - Robert G. Wilson v, Jul 27 2006

Examples

			a(20) = 1 + 8 = 9 because 20 is written as 18 base 12.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• 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, Duodecimal.
• Eric Weisstein's World of Mathematics, Digit Sum.

Crossrefs

Cf. A000120, A007953, A064459, A138530.

Programs

Haskell

a053832 n = q 0 $ divMod n 12 where
   q r (0, d) = r + d
   q r (m, d) = q (r + d) $ divMod m 12
-- Reinhard Zumkeller, May 15 2011

Mathematica

Table[Plus @@ IntegerDigits[n, 12], {n, 0, 85}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 11}]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)

PARI

a(n)=if(n<1,0,if(n%12,a(n-1)+1,a(n/12)))

Formula

From Benoit Cloitre, Dec 19 2002: (Start)

a(0) = 0, a(12n+i) = a(n)+i for 0 <= i <= 11.

a(n) = n-11*(Sum_{k>0} floor(n/12^k)) = n-11*A064459(n). (End)

a(n) = A138530(n,12) for n > 11. - Reinhard Zumkeller, Mar 26 2008

Sum_{n>=1} a(n)/(n*(n+1)) = 12*log(12)/11 (Shallit, 1984). - Amiram Eldar, Jun 03 2021