A053829 - OEIS

%I #50 Jul 10 2022 09:43:00

%S 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,

%T 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,

%U 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

%N Sum of digits of (n written in base 8).

%C 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

%H Reinhard Zumkeller, <a href="/A053829/b053829.txt">Table of n, a(n) for n = 0..10000</a>

%H Jeffrey O. Shallit, <a href="http://www.jstor.org/stable/2322179">Problem 6450</a>, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; <a href="http://www.jstor.org/stable/2322523">Two series, solution to Problem 6450</a>, ibid., Vol. 92, No. 7 (1985), pp. 513-514.

%H Robert Walker, <a href="http://robertinventor.com/ftswiki/Self_Similar_Sloth_Canon_Number_Sequences">Self Similar Sloth Canon Number Sequences</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitSum.html">Digit Sum</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Octal.html">Octal</a>.

%F From _Benoit Cloitre_, Dec 19 2002: (Start)

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

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

%F a(n) = A138530(n,8) for n > 7. - _Reinhard Zumkeller_, Mar 26 2008

%F a(n) = Sum_k>=0 {A031045(n,k)}. - _Philippe Deléham_, Oct 21 2011

%F a(0) = 0; a(n) = a(n - 8^floor(log_8(n))) + 1. - _Ilya Gutkovskiy_, Aug 24 2019

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

%e a(20)=2+4=6 because 20 is written as 24 base 8.

%e From _Omar E. Pol_, Feb 21 2010: (Start)

%e It appears that this can be written as a triangle (See the conjecture in the entry A000120):

%e 0,

%e 1,2,3,4,5,6,7,

%e 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,

%e 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...

%e where the rows converge to A173528. (End)

%t Table[Plus @@ IntegerDigits[n, 8], {n, 0, 95}] (* or *)

%t Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 7}]] &, {0}, 4] (* _Robert G. Wilson v_, Jul 27 2006 *)

%o (PARI) a(n)=if(n<1,0,if(n%8,a(n-1)+1,a(n/8)))

%o (PARI) a(n) = sumdigits(n, 8); \\ _Michel Marcus_, Jul 10 2022

%o (Haskell)

%o a053829 n = q 0 $ divMod n 8 where

%o    q r (0, d) = r + d

%o    q r (m, d) = q (r + d) $ divMod m 8

%o -- _Reinhard Zumkeller_, May 15 2011

%o (Python)

%o def A053829(n): return sum(int(d) for d in oct(n)[2:]) # _Chai Wah Wu_, Jul 09 2022

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

%Y Cf. A173528. - _Omar E. Pol_, Feb 21 2010

%K base,nonn

%O 0,3

%A _Henry Bottomley_, Mar 28 2000