A231669 -id:A231669 - cp's OEIS frontend

A053824 Sum of digits of (n written in base 5).

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, 6, 7, 8, 5, 6, 7, 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

Offset: 0

Henry Bottomley, Mar 28 2000

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

			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)

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.

Cf. A231668, A231669, A231670, A231671, A138530.
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

a053824 0 = 0
a053824 x = a053824 x' + d  where (x', d) = divMod x 5
-- Reinhard Zumkeller, Jan 31 2014

[&+Intseq(n, 5):n in [0..100]]; // Marius A. Burtea, Aug 24 2019

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]

PARI
```
a(n)=if(n<1,0,if(n%5,a(n-1)+1,a(n/5)))
```

a(n) = sumdigits(n, 5); \\ Michel Marcus, Aug 24 2019

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

A231668 a(n) = Sum_{i=0..n} digsum_5(i), where digsum_5(i) = A053824(i).

Original entry on oeis.org

0, 1, 3, 6, 10, 11, 13, 16, 20, 25, 27, 30, 34, 39, 45, 48, 52, 57, 63, 70, 74, 79, 85, 92, 100, 101, 103, 106, 110, 115, 117, 120, 124, 129, 135, 138, 142, 147, 153, 160, 164, 169, 175, 182, 190, 195, 201, 208, 216, 225, 227, 230, 234, 239, 245, 248, 252, 257, 263, 270, 274, 279, 285, 292, 300, 305, 311, 318, 326, 335, 341, 348, 356, 365, 375, 378, 382, 387, 393, 400

Offset: 0

N. J. A. Sloane, Nov 13 2013

Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Cambridge University Press, 2003, p. 94.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Jean Coquet, Power sums of digital sums, J. Number Theory, Vol. 22, No. 2 (1986), pp. 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), pp. 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
Hsien-Kuei Hwang, Svante Janson and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, Vol. 13, No. 4 (2017), Article #47; ResearchGate link; preprint, 2016.
J.-L. Mauclaire and Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 6 (1983), pp. 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 9 (1983), pp. 441-444.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.

Cf. A053824, A231669, A231670, A231671.

a[n_] := Plus @@ IntegerDigits[n, 5]; Accumulate @ Array[a, 80, 0] (* Amiram Eldar, Dec 09 2021 *)

a(n) = sum(i=0, n, sumdigits(i, 5)); \\ Michel Marcus, Sep 20 2017

a(n) ~ 2*n*log(n)/log(5). - Amiram Eldar, Dec 09 2021

A231670 a(n) = Sum_{i=0..n} digsum_5(i)^3, where digsum_5(i) = A053824(i).

Original entry on oeis.org

0, 1, 9, 36, 100, 101, 109, 136, 200, 325, 333, 360, 424, 549, 765, 792, 856, 981, 1197, 1540, 1604, 1729, 1945, 2288, 2800, 2801, 2809, 2836, 2900, 3025, 3033, 3060, 3124, 3249, 3465, 3492, 3556, 3681, 3897, 4240, 4304, 4429, 4645, 4988, 5500, 5625, 5841, 6184, 6696, 7425, 7433, 7460, 7524, 7649, 7865, 7892, 7956, 8081, 8297, 8640, 8704, 8829, 9045, 9388, 9900, 10025

Offset: 0

N. J. A. Sloane, Nov 13 2013

Grabner, P. J.; Kirschenhofer, P.; Prodinger, H.; Tichy, R. F.; On the moments of the sum-of-digits function. Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
J.-L. Mauclaire and Leo Murata, On q-additive functions, I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions, II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A053824, A231668, A231669, A231671.

Accumulate[f[n_]:=n - 4 Sum[Floor[n/5^k], {k, n}]; Array[f, 100, 0]^3] (* Vincenzo Librandi, Sep 04 2016 *)

A231671 a(n) = Sum_{i=0..n} digsum_5(i)^4, where digsum_5(i) = A053824(i).

Original entry on oeis.org

0, 1, 17, 98, 354, 355, 371, 452, 708, 1333, 1349, 1430, 1686, 2311, 3607, 3688, 3944, 4569, 5865, 8266, 8522, 9147, 10443, 12844, 16940, 16941, 16957, 17038, 17294, 17919, 17935, 18016, 18272, 18897, 20193, 20274, 20530, 21155, 22451, 24852, 25108, 25733, 27029, 29430, 33526, 34151, 35447, 37848, 41944, 48505, 48521, 48602, 48858, 49483, 50779, 50860, 51116, 51741

Offset: 0

N. J. A. Sloane, Nov 13 2013

Grabner, P. J.; Kirschenhofer, P.; Prodinger, H.; Tichy, R. F.; On the moments of the sum-of-digits function. Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
J.-L. Mauclaire and Leo Murata, On q-additive functions, I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions, II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A053824, A231668, A231669, A231670.

Accumulate[f[n_]:=n - 4 Sum[Floor[n/5^k], {k, n}]; Array[f, 100, 0]^4] (* Vincenzo Librandi, Sep 04 2016 *)

cp's OEIS Frontend

A053824 Sum of digits of (n written in base 5).

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A231668 a(n) = Sum_{i=0..n} digsum_5(i), where digsum_5(i) = A053824(i).

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A231670 a(n) = Sum_{i=0..n} digsum_5(i)^3, where digsum_5(i) = A053824(i).

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A231671 a(n) = Sum_{i=0..n} digsum_5(i)^4, where digsum_5(i) = A053824(i).

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