A053829
Sum of digits of (n written in base 8).
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 10, 4, 5, 6, 7, 8, 9, 10, 11, 5, 6, 7, 8, 9, 10, 11, 12, 6, 7, 8, 9, 10, 11, 12, 13, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 10, 4, 5, 6, 7, 8, 9, 10
Offset: 0
a(20)=2+4=6 because 20 is written as 24 base 8.
From _Omar E. Pol_, Feb 21 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,
1,2,3,4,5,6,7,8,2,3,4,5,6,7,8,9,3,4,5,6,7,8,9,10,4,5,6,7,8,9,10,11,5,6,7,8,9,10,11,12,6,7,8,9,10,11,12,13,7,8,9,10,11,12,13,14,
1,2,3,4,5,6,7,8,2,3,4,5,6,7,8,9,3,4,5,6,7,8,9,10,4,5,6,7,8,9,10...
where the rows converge to A173528. (End)
- Reinhard Zumkeller, 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.
- Eric Weisstein's World of Mathematics, Octal.
-
a053829 n = q 0 $ divMod n 8 where
q r (0, d) = r + d
q r (m, d) = q (r + d) $ divMod m 8
-- Reinhard Zumkeller, May 15 2011
-
Table[Plus @@ IntegerDigits[n, 8], {n, 0, 95}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 7}]] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)
-
a(n)=if(n<1,0,if(n%8,a(n-1)+1,a(n/8)))
-
a(n) = sumdigits(n, 8); \\ Michel Marcus, Jul 10 2022
-
def A053829(n): return sum(int(d) for d in oct(n)[2:]) # Chai Wah Wu, Jul 09 2022
A231680
a(n) = Sum_{i=0..n} digsum_8(i), where digsum_8(i) = A053829(i).
Original entry on oeis.org
0, 1, 3, 6, 10, 15, 21, 28, 29, 31, 34, 38, 43, 49, 56, 64, 66, 69, 73, 78, 84, 91, 99, 108, 111, 115, 120, 126, 133, 141, 150, 160, 164, 169, 175, 182, 190, 199, 209, 220, 225, 231, 238, 246, 255, 265, 276, 288, 294, 301, 309, 318, 328, 339, 351, 364, 371, 379, 388, 398, 409, 421, 434, 448, 449, 451, 454, 458, 463, 469, 476, 484, 486, 489, 493, 498, 504, 511, 519, 528
Offset: 0
- 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.
-
a[n_] := Plus @@ IntegerDigits[n, 8]; Accumulate @ Array[a, 80, 0] (* Amiram Eldar, Dec 09 2021 *)
-
a(n) = sum(i=0, n, sumdigits(i, 8)); \\ Michel Marcus, Sep 20 2017
A231682
a(n) = Sum_{i=0..n} digsum_8(i)^3, where digsum_8(i) = A053829(i).
Original entry on oeis.org
0, 1, 9, 36, 100, 225, 441, 784, 785, 793, 820, 884, 1009, 1225, 1568, 2080, 2088, 2115, 2179, 2304, 2520, 2863, 3375, 4104, 4131, 4195, 4320, 4536, 4879, 5391, 6120, 7120, 7184, 7309, 7525, 7868, 8380, 9109, 10109, 11440, 11565, 11781, 12124, 12636, 13365, 14365, 15696, 17424, 17640, 17983, 18495, 19224, 20224, 21555, 23283, 25480, 25823, 26335, 27064, 28064, 29395
Offset: 0
- 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.
-
a(n) = sum(i=0, n, sumdigits(i, 8)^3); \\ Michel Marcus, Sep 20 2017
A231683
a(n) = Sum_{i=0..n} digsum_8(i)^4, where digsum_8(i) = A053829(i).
Original entry on oeis.org
0, 1, 17, 98, 354, 979, 2275, 4676, 4677, 4693, 4774, 5030, 5655, 6951, 9352, 13448, 13464, 13545, 13801, 14426, 15722, 18123, 22219, 28780, 28861, 29117, 29742, 31038, 33439, 37535, 44096, 54096, 54352, 54977, 56273, 58674, 62770, 69331, 79331, 93972, 94597, 95893, 98294, 102390, 108951, 118951, 133592, 154328, 155624, 158025, 162121, 168682, 178682, 193323, 214059
Offset: 0
- 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.
-
a(n) = sum(i=0, n, sumdigits(i, 8)^4); \\ Michel Marcus, Sep 20 2017
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