OFFSET
0,3
COMMENTS
Also the fixed point of the morphism 0->{0,1,2,3,4}, 1->{1,2,3,4,5}, 2->{2,3,4,5,6}, etc. - Robert G. Wilson v, Jul 27 2006
LINKS
Tanar Ulric, Table of n, a(n) for n = 0..10000 (terms 0..3125=5^5 from Reinhard Zumkeller).
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.
FORMULA
From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(5n+i) = a(n) + i for 0 <= i <= 4;
a(n) = n - 4*Sum_{k>=1} floor(n/5^k) = n - 4*A027868(n). (End)
a(n) = A138530(n,5) for n > 4. - Reinhard Zumkeller, Mar 26 2008
If i >= 2, a(2^i) mod 4 = 0. - Washington Bomfim, Jan 01 2011
a(n) = Sum_{k>=0} A031235(n,k). - Philippe Deléham, Oct 21 2011
a(0) = 0; a(n) = a(n - 5^floor(log_5(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 5*log(5)/4 (Shallit, 1984). - Amiram Eldar, Jun 03 2021
EXAMPLE
a(20) = 4 + 0 = 4 because 20 is written as 40 in base 5.
From Omar E. Pol, Feb 21 2010: (Start)
It appears that this can be written as a triangle:
0,
1,2,3,4,
1,2,3,4,5,2,3,4,5,6,3,4,5,6,7,4,5,6,7,8,
1,2,3,4,5,2,3,4,5,6,3,4,5,6,7,4,5,6,7,8,5,6,7,8,9,2,3,4,5,6,3,4,5,6,7,4,5,...
See the conjecture in the entry A000120. (End)
MATHEMATICA
Table[Plus @@ IntegerDigits[n, 5], {n, 0, 100}] (* or *)
Nest[Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 4}]] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)
f[n_] := n - 4 Sum[Floor[n/5^k], {k, n}]; Array[f, 103, 0]
PROG
(PARI) a(n)=if(n<1, 0, if(n%5, a(n-1)+1, a(n/5)))
(PARI) a(n) = sumdigits(n, 5); \\ Michel Marcus, Aug 24 2019
(Haskell)
a053824 0 = 0
a053824 x = a053824 x' + d where (x', d) = divMod x 5
-- Reinhard Zumkeller, Jan 31 2014
(Magma) [&+Intseq(n, 5):n in [0..100]]; // Marius A. Burtea, Aug 24 2019
CROSSREFS
Sum of digits of n written in bases 2-16: A000120, A053735, A053737, this sequence, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Cf. A173525. - Omar E. Pol, Feb 21 2010
Cf. A173670 (last nonzero decimal digit of (10^n)!). - Washington Bomfim, Jan 01 2011
KEYWORD
AUTHOR
Henry Bottomley, Mar 28 2000
STATUS
approved