A231687 -id:A231687 - cp's OEIS frontend

A053830 Sum of digits of (n written in base 9).

0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8}, 1->{1,2,3,4,5,6,7,8,9}, 2->{2,3,4,5,6,7,8,9,10}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 2+2 = 4 because 20 is written as 22 base 9.
From _Omar E. Pol_, Feb 23 2010: (Start)
It appears that this can be written as a triangle (see the conjecture in the entry A000120):
0;
1,2,3,4,5,6,7,8;
1,2,3,4,5,6,7,8,9,2,3,4,5,6,7,8,9,10,3,4,5,6,7,8,9,10,11,4,5,6,7,8,9,10,11,...
where the rows converge to A173529. (End)

Indranil Ghosh, Table of n, a(n) for n = 0..59049
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. A000120, A007953, A053735, A053737, A053824, A053828, A231684, A231685, A231686, A231687.

Cf. A173529. - Omar E. Pol, Feb 23 2010

[&+Intseq(n, 9):n in [0..100]]; // Marius A. Burtea, Aug 24 2019

Table[Plus @@ IntegerDigits[n, 9], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 8}]] &, {0}, 3] (* Robert G. Wilson v, Jul 27 2006 *)

PARI
```
a(n)=if(n<1,0,if(n%9,a(n-1)+1,a(n/9)))
```

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(9n+i) = a(n) + i for 0 <= i <= 8;
a(n) = n - 8*Sum_{k>=1} floor(n/9^k) = n - 8*A054898(n). (End)
a(n) = A138530(n,9) for n > 8. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_{k>=0} A031087(n,k). - Philippe Deléham, Oct 21 2011
a(0) = 0; a(n) = a(n - 9^floor(log_9(n))) + 1. - Ilya Gutkovskiy, Aug 24 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 9*log(9)/8 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A231684 a(n) = Sum_{i=0..n} digsum_9(i), where digsum_9(i) = A053830(i).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 37, 39, 42, 46, 51, 57, 64, 72, 81, 83, 86, 90, 95, 101, 108, 116, 125, 135, 138, 142, 147, 153, 160, 168, 177, 187, 198, 202, 207, 213, 220, 228, 237, 247, 258, 270, 275, 281, 288, 296, 305, 315, 326, 338, 351, 357, 364, 372, 381, 391, 402, 414, 427, 441, 448, 456, 465, 475, 486, 498, 511, 525, 540, 548, 557, 567, 578, 590, 603, 617, 632

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. A053830, A231685, A231686, A231687.

a[n_] := Plus @@ IntegerDigits[n, 9]; Accumulate @ Array[a, 80, 0] (* Amiram Eldar, Dec 09 2021 *)

a(n) = sum(i=0, n, sumdigits(i, 9)); \\ Michel Marcus, Sep 20 2017

a(n) ~ 2*n*log(n)/log(3). - Amiram Eldar, Dec 09 2021

A231685 a(n) = Sum_{i=0..n} digsum_9(i)^2, where digsum_9(i) = A053830(i).

Original entry on oeis.org

0, 1, 5, 14, 30, 55, 91, 140, 204, 205, 209, 218, 234, 259, 295, 344, 408, 489, 493, 502, 518, 543, 579, 628, 692, 773, 873, 882, 898, 923, 959, 1008, 1072, 1153, 1253, 1374, 1390, 1415, 1451, 1500, 1564, 1645, 1745, 1866, 2010, 2035, 2071, 2120, 2184, 2265, 2365, 2486, 2630, 2799, 2835, 2884, 2948, 3029, 3129, 3250, 3394, 3563, 3759, 3808, 3872, 3953, 4053, 4174, 4318

Offset: 0

N. J. A. Sloane, Nov 13 2013

Partial sums of ((the total of the digits of i in base 9) squared). - Harvey P. Dale, Nov 26 2013

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Jean Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger, R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
J.-L. Mauclaire, Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire, 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. A053830, A231684, A231686, A231687.

Accumulate[Table[Total[IntegerDigits[n,9]]^2,{n,0,100}]] (* Harvey P. Dale, Nov 26 2013 *)

a(n) = sum(i=0, n, sumdigits(i, 9)^2); \\ Michel Marcus, Sep 20 2017

A231686 a(n) = Sum_{i=0..n} digsum_9(i)^3, where digsum_9(i) = A053830(i).

Original entry on oeis.org

0, 1, 9, 36, 100, 225, 441, 784, 1296, 1297, 1305, 1332, 1396, 1521, 1737, 2080, 2592, 3321, 3329, 3356, 3420, 3545, 3761, 4104, 4616, 5345, 6345, 6372, 6436, 6561, 6777, 7120, 7632, 8361, 9361, 10692, 10756, 10881, 11097, 11440, 11952, 12681, 13681, 15012, 16740, 16865, 17081, 17424, 17936, 18665, 19665, 20996, 22724, 24921, 25137, 25480, 25992, 26721, 27721, 29052

Offset: 0

N. J. A. Sloane, Nov 13 2013

Jean Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger, R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
J.-L. Mauclaire, Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire, 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. A053830, A231684, A231685, A231687.

Accumulate[Table[Total[IntegerDigits[n,9]]^3,{n,0,60}]] (* Harvey P. Dale, Dec 22 2020 *)

a(n) = sum(i=0, n, sumdigits(i, 9)^3); \\ Michel Marcus, Sep 20 2017

cp's OEIS Frontend

A053830 Sum of digits of (n written in base 9).

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A231684 a(n) = Sum_{i=0..n} digsum_9(i), where digsum_9(i) = A053830(i).

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A231685 a(n) = Sum_{i=0..n} digsum_9(i)^2, where digsum_9(i) = A053830(i).

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Author

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Comments

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Mathematica

PARI

A231686 a(n) = Sum_{i=0..n} digsum_9(i)^3, where digsum_9(i) = A053830(i).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI