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%I A053824 #76 Feb 15 2024 18:57:48
%S A053824 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,
%T A053824 6,3,4,5,6,7,4,5,6,7,8,5,6,7,8,9,2,3,4,5,6,3,4,5,6,7,4,5,6,7,8,5,6,7,
%U A053824 8,9,6,7,8,9,10,3,4,5,6,7,4,5,6,7,8,5,6,7,8,9,6,7,8,9,10,7,8,9,10,11,4,5,6
%N A053824 Sum of digits of (n written in base 5).
%C A053824 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
%H A053824 Tanar Ulric, <a href="/A053824/b053824.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..3125=5^5 from Reinhard Zumkeller).
%H A053824 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 A053824 Robert Walker, <a href="http://robertinventor.com/ftswiki/Self_Similar_Sloth_Canon_Number_Sequences">Self Similar Sloth Canon Number Sequences</a>.
%H A053824 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitSum.html">Digit Sum</a>.
%F A053824 From _Benoit Cloitre_, Dec 19 2002: (Start)
%F A053824 a(0) = 0, a(5n+i) = a(n) + i for 0 <= i <= 4;
%F A053824 a(n) = n - 4*Sum_{k>=1} floor(n/5^k) = n - 4*A027868(n). (End)
%F A053824 a(n) = A138530(n,5) for n > 4. - _Reinhard Zumkeller_, Mar 26 2008
%F A053824 If i >= 2, a(2^i) mod 4 = 0. - _Washington Bomfim_, Jan 01 2011
%F A053824 a(n) = Sum_{k>=0} A031235(n,k). - _Philippe Deléham_, Oct 21 2011
%F A053824 a(0) = 0; a(n) = a(n - 5^floor(log_5(n))) + 1. - _Ilya Gutkovskiy_, Aug 23 2019
%F A053824 Sum_{n>=1} a(n)/(n*(n+1)) = 5*log(5)/4 (Shallit, 1984). - _Amiram Eldar_, Jun 03 2021
%e A053824 a(20) = 4 + 0 = 4 because 20 is written as 40 in base 5.
%e A053824 From _Omar E. Pol_, Feb 21 2010: (Start)
%e A053824 It appears that this can be written as a triangle:
%e A053824   0,
%e A053824   1,2,3,4,
%e A053824   1,2,3,4,5,2,3,4,5,6,3,4,5,6,7,4,5,6,7,8,
%e A053824   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,...
%e A053824 See the conjecture in the entry A000120. (End)
%t A053824 Table[Plus @@ IntegerDigits[n, 5], {n, 0, 100}] (* or *)
%t A053824 Nest[Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 4}]] &, {0}, 4] (* _Robert G. Wilson v_, Jul 27 2006 *)
%t A053824 f[n_] := n - 4 Sum[Floor[n/5^k], {k, n}]; Array[f, 103, 0]
%o A053824 (PARI) a(n)=if(n<1,0,if(n%5,a(n-1)+1,a(n/5)))
%o A053824 (PARI) a(n) = sumdigits(n, 5); \\ _Michel Marcus_, Aug 24 2019
%o A053824 (Haskell)
%o A053824 a053824 0 = 0
%o A053824 a053824 x = a053824 x' + d  where (x', d) = divMod x 5
%o A053824 -- _Reinhard Zumkeller_, Jan 31 2014
%o A053824 (Magma) [&+Intseq(n, 5):n in [0..100]]; // _Marius A. Burtea_, Aug 24 2019
%Y A053824 Cf. A231668, A231669, A231670, A231671, A138530.
%Y A053824 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.
%Y A053824 Cf. A173525. - _Omar E. Pol_, Feb 21 2010
%Y A053824 Cf. A173670 (last nonzero decimal digit of (10^n)!). - _Washington Bomfim_, Jan 01 2011
%K A053824 base,nonn,look
%O A053824 0,3
%A A053824 _Henry Bottomley_, Mar 28 2000

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