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

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

Author

N. J. A. Sloane, Nov 13 2013

Keywords

• nonn
• base

References

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

Links

• 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.

Crossrefs

Cf. A053827, A231673, A231674, A231675.

Programs

Mathematica

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

PARI

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

Formula

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