A053829 Sum of digits of (n written in base 8).

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

Offset

0,3

Comments

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

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, Digit Sum.

Eric Weisstein's World of Mathematics, Octal.

Formula

From Benoit Cloitre, Dec 19 2002: (Start)

a(0) = 0, a(8n+i) = a(n)+i for 0 <= i <= 7.

a(n) = n-7*(Sum_{k>0} floor(n/8^k)) = n-7*A054897(n). (End)

a(n) = A138530(n,8) for n > 7. - Reinhard Zumkeller, Mar 26 2008

a(n) = Sum_k>=0 {A031045(n,k)}. - Philippe Deléham, Oct 21 2011

a(0) = 0; a(n) = a(n - 8^floor(log_8(n))) + 1. - Ilya Gutkovskiy, Aug 24 2019

Sum_{n>=1} a(n)/(n*(n+1)) = 8*log(8)/7 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

Example

a(20)=2+4=6 because 20 is written as 24 base 8.
From Omar E. Pol, Feb 21 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,
1,2,3,4,5,6,7,8,2,3,4,5,6,7,8,9,3,4,5,6,7,8,9,10,4,5,6,7,8,9,10,11,5,6,7,8,9,10,11,12,6,7,8,9,10,11,12,13,7,8,9,10,11,12,13,14,
1,2,3,4,5,6,7,8,2,3,4,5,6,7,8,9,3,4,5,6,7,8,9,10,4,5,6,7,8,9,10...
where the rows converge to A173528. (End)

Mathematica

Table[Plus @@ IntegerDigits[n, 8], {n, 0, 95}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 7}]] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)

Prog

(PARI) a(n)=if(n<1, 0, if(n%8, a(n-1)+1, a(n/8)))
(PARI) a(n) = sumdigits(n, 8); \\ Michel Marcus, Jul 10 2022
(Haskell)
a053829 n = q 0 $ divMod n 8 where
   q r (0, d) = r + d
   q r (m, d) = q (r + d) $ divMod m 8
-- Reinhard Zumkeller, May 15 2011
(Python)
def A053829(n): return sum(int(d) for d in oct(n)[2:]) # Chai Wah Wu, Jul 09 2022

Crossrefs

Cf. A000120, A007953, A231680, A231681, A231682, A231683.

Cf. A173528. - Omar E. Pol, Feb 21 2010

Sequence in context: A338458 A002376 A055401 * A033928 A194754 A167972

Adjacent sequences: A053826 A053827 A053828 * A053830 A053831 A053832

Keyword

base,nonn

Author

Henry Bottomley, Mar 28 2000

Status

approved