A231675 -id:A231675 - cp's OEIS frontend

A053827 Sum of digits of (n written in base 6).

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

Offset: 0

Henry Bottomley, Mar 28 2000

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

Sum of six consecutive terms is (15,21,27,33,39,45; 21,27,33,39,45,51; 27,33,39,45,51,57; and so on). - Vincenzo Librandi, Aug 02 2010

			a(20)=3+2=5 because 20 is written as 32 base 6.
From _Omar E. Pol_, Feb 21 2010: (Start)
It appears that this can be written as a triangle :
  0,
  1,2,3,4,5,
  1,2,3,4,5,6,2,3,4,5,6,7,3,4,5,6,7,8,4,5,6,7,8,9,5,6,7,8,9,10,
  1,2,3,4,5,6,2,3,4,5,6,7,3,4,5,6,7,8,4,5,6,7,8,9,5,6,7,8,9,10,6,7,8,9,10,11,2...
where the rows converge to A173526.
See the conjecture in the entry A000120. (End)

Indranil Ghosh, 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.

Cf. A231672, A231673, A231674, A231675, A138530.
Sum of digits of n written in bases 2-16: A000120, A053735, A053737, A053824, this sequence, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Cf. A173526. - Omar E. Pol, Feb 21 2010

[&+Intseq(n,6):n in [0..105]]; // Marius A. Burtea, Aug 24 2019

Table[Plus @@ IntegerDigits[n, 6], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 5}]] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)

PARI
```
a(n)=if(n<1,0,if(n%6,a(n-1)+1,a(n/6)))
```

a(n) = sumdigits(n, 6); \\ Michel Marcus, Aug 24 2019

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(6n+i) = a(n)+i for 0 <= i <= 5.
a(n) = n-5*(Sum_{k>0} floor(n/6^k)) = n-5*A054895(n). (End)
a(n) = A138530(n,6) for n > 5. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_{k>=0} A030567(n,k). - Philippe Deléham, Oct 21 2011
a(0) = 0; a(n) = a(n - 6^floor(log_6(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 6*log(6)/5 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A231672 a(n) = Sum_{i=0..n} digsum_6(i), where digsum_6(i) = A053827(i).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 16, 18, 21, 25, 30, 36, 38, 41, 45, 50, 56, 63, 66, 70, 75, 81, 88, 96, 100, 105, 111, 118, 126, 135, 140, 146, 153, 161, 170, 180, 181, 183, 186, 190, 195, 201, 203, 206, 210, 215, 221, 228, 231, 235, 240, 246, 253, 261, 265, 270, 276, 283, 291, 300, 305, 311, 318, 326, 335, 345, 351, 358, 366, 375, 385, 396, 398, 401, 405, 410, 416, 423, 426, 430

Offset: 0

N. J. A. Sloane, Nov 13 2013

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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. A053827, A231673, A231674, A231675.

Accumulate[f[n_]:=n - 5 Sum[Floor[n/6^k], {k, n}]; Array[f, 100, 0]] (* Vincenzo Librandi, Sep 04 2016 *)

a(n) = sum(i=0, n, sumdigits(i, 6)); \\ Michel Marcus, Dec 09 2021

a(n) ~ 5*n*log(n)/(2*log(6)). - Amiram Eldar, Dec 09 2021

A231673 a(n) = Sum_{i=0..n} digsum_6(i)^2, where digsum_6(i) = A053827(i).

Original entry on oeis.org

0, 1, 5, 14, 30, 55, 56, 60, 69, 85, 110, 146, 150, 159, 175, 200, 236, 285, 294, 310, 335, 371, 420, 484, 500, 525, 561, 610, 674, 755, 780, 816, 865, 929, 1010, 1110, 1111, 1115, 1124, 1140, 1165, 1201, 1205, 1214, 1230, 1255, 1291, 1340, 1349, 1365, 1390, 1426, 1475, 1539, 1555, 1580, 1616, 1665, 1729, 1810, 1835, 1871, 1920, 1984, 2065, 2165, 2201, 2250, 2314, 2395

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. A053827, A231672, A231674, A231675.

Accumulate[f[n_]:=n - 5 Sum[Floor[n/6^k], {k, n}]; Array[f, 100, 0]^2] (* Vincenzo Librandi, Sep 04 2016 *)

A231674 a(n) = Sum_{i=0..n} digsum_6(i)^3, where digsum_6(i) = A053827(i).

Original entry on oeis.org

0, 1, 9, 36, 100, 225, 226, 234, 261, 325, 450, 666, 674, 701, 765, 890, 1106, 1449, 1476, 1540, 1665, 1881, 2224, 2736, 2800, 2925, 3141, 3484, 3996, 4725, 4850, 5066, 5409, 5921, 6650, 7650, 7651, 7659, 7686, 7750, 7875, 8091, 8099, 8126, 8190, 8315, 8531, 8874, 8901, 8965, 9090, 9306, 9649, 10161, 10225, 10350, 10566, 10909, 11421, 12150, 12275, 12491, 12834, 13346

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. A053827, A231672, A231673, A231675.

Accumulate[f[n_]:=n - 5Sum[Floor[n/6^k], {k, n}]; Array[f, 100, 0]^3] (* Vincenzo Librandi, Sep 04 2016 *)

cp's OEIS Frontend

A053827 Sum of digits of (n written in base 6).

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A231672 a(n) = Sum_{i=0..n} digsum_6(i), where digsum_6(i) = A053827(i).

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A231673 a(n) = Sum_{i=0..n} digsum_6(i)^2, where digsum_6(i) = A053827(i).

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A231674 a(n) = Sum_{i=0..n} digsum_6(i)^3, where digsum_6(i) = A053827(i).

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