A073636 -id:A073636 - cp's OEIS frontend

A004164 Sum of digits of n^3.

0, 1, 8, 9, 10, 8, 9, 10, 8, 18, 1, 8, 18, 19, 17, 18, 19, 17, 18, 28, 8, 18, 19, 17, 18, 19, 26, 27, 19, 26, 9, 28, 26, 27, 19, 26, 27, 19, 26, 27, 10, 26, 27, 28, 26, 18, 28, 17, 18, 28, 8, 18, 19, 35, 27, 28, 26, 27, 19, 26, 9, 28, 26, 18, 19, 26, 36, 19

Offset: 0

N. J. A. Sloane

For the digital root of n^3 see A073636.

The greedy inverse is 1, -1, -1, -1, -1, -1, -1, 2, 3, 4, -1, -1, -1, -1, -1, -1, 14, 9, 13, -1, -1, .. where -1 means the inverse does not exist. Essentially provided by A067177. - R. J. Mathar, Jul 19 2024

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Michael Drmota, Christian Mauduit, Joël Rivat, The sum-of-digits function of polynomial sequences, J. Lond. Math. Soc. (2) 84(2011), no. 1, 81--102. MR2819691 (2012f:11193).

Cf. A004159, A007953.

A004164 := proc(n) return add(d, d=convert(n^3,base,10)): end: seq(A004164(n),n=0..100); # Nathaniel Johnston, May 05 2011

DigitSum[Range[0, 100]^3] (* Paolo Xausa, Jul 03 2024 *)

a(n) = sumdigits(n^3); \\ Michel Marcus, Jul 07 2019

A016791 a(n) = (3*n + 2)^3.

Original entry on oeis.org

8, 125, 512, 1331, 2744, 4913, 8000, 12167, 17576, 24389, 32768, 42875, 54872, 68921, 85184, 103823, 125000, 148877, 175616, 205379, 238328, 274625, 314432, 357911, 405224, 456533, 512000, 571787, 636056, 704969, 778688, 857375, 941192, 1030301, 1124864, 1225043

Offset: 0

N. J. A. Sloane

Also the perfect cubes with digital root 8. [Proof: perfect cubes are either of the form (3n)^3 or of the form (3n+1)^3 or of the form (3n+2)^3. These subsets have digital root 9, 1 and 8 respectively.] - R. J. Mathar, Oct 02 2008

			a(4) = (3*4 + 2)^3 = 2744.
a(8) = (3*8 + 2)^3 = 17576.

Amarnath Murthy, Fabricating a perfect cube with a given valid digit sum (to be published)

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A054966, A016779, A016789, A073636.

(3*Range[0,40]+2)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{8,125,512,1331},40] (* Harvey P. Dale, Feb 20 2013 *)

a(n) = { (3*n + 2)^3 } \\ Harry J. Smith, Jul 18 2009

a(n) = A016789(n)^3. - Nathaniel Johnston, May 04 2011
G.f.: (8 + 93*x + 60*x^2 + x^3)/(1 - 4*x + 6*x^2 - 4*x^3 + x^4). - Colin Barker, Jan 02 2012
a(0)=8, a(1)=125, a(2)=512, a(3)=1331, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Feb 20 2013
Sum_{n>=0} 1/a(n) = -2*Pi^3 / (81*sqrt(3)) + 13*zeta(3)/27. - Amiram Eldar, Oct 02 2020

More terms from Harry J. Smith, Jul 18 2009

First digital root in proof in comment line corrected. - Ant King, May 01 2013

A353109 Array read by antidiagonals: A(n, k) is the digital root of n*k with n >= 0 and k >= 0.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Apr 24 2022

			The array begins:
    0, 0, 0, 0, 0, 0, 0, 0, ...
    0, 1, 2, 3, 4, 5, 6, 7, ...
    0, 2, 4, 6, 8, 1, 3, 5, ...
    0, 3, 6, 9, 3, 6, 9, 3, ...
    0, 4, 8, 3, 7, 2, 6, 1, ...
    0, 5, 1, 6, 2, 7, 3, 8, ...
    0, 6, 3, 9, 6, 3, 9, 6, ...
    0, 7, 5, 3, 1, 8, 6, 4, ...
    ...

Cf. A003991, A004247, A010888, A056992 (diagonal), A073636, A139413, A180592, A180593, A180594, A180595, A180596, A180597, A180598, A180599, A303296, A336225, A353128 (antidiagonal sums), A353933, A353974 (partial sum of the main diagonal).

A[i_,j_]:=If[i*j==0,0,1+Mod[i*j-1,9]];Flatten[Table[A[n-k,k],{n,0,12},{k,0,n}]]

T(n,k) = if (n && k, (n*k-1)%9+1, 0); \\ Michel Marcus, May 12 2022

A(n, k) = A010888(A004247(n, k)).

A(n, k) = A010888(A003991(n, k)) for n*k > 0.

A251754 Digital root of A027444(n) = n + n^2 + n^3, n>=1. Repeat(3, 5, 3, 3, 2, 6, 3, 8, 9).

Original entry on oeis.org

3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3

Offset: 1

Peter M. Chema, Dec 07 2014

Periodic with cycle of length 9: {3, 5, 3, 3, 2, 6, 3, 8, 9}.

a(n) also arises from the decimal expansion of 117775463/333333333 = 0.repeat(353326389).

			For a(11) = 5 because 11+11^2+11^3 = 1463, and 1+4+6+3 = 14.  Result is 5, which is the digital root of 14.

Eric Weisstein's World of Mathematics, Digital Root.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A027444, A010888, A056992, A073636.

&cat [[3,5,3,3,2,6,3,8,9]^^10]; // Vincenzo Librandi, Jul 18 2016

PadRight[{}, 120, {3, 5, 3, 3, 2, 6, 3, 8, 9}] (* Vincenzo Librandi, Jul 18 2016 *)

a(n) = sum of digits of (n+n^2+n^3), reduced to digital root.
a(n) = A010888(A027444(n)), and sequence may start at n=0.
a(n) = A010888(A010888(n) + A056992(n) + A073636(n)).
G.f.: x*(9*x^8 + 8*x^7 + 3*x^6 + 6*x^5 + 2*x^4 + 3*x^3 + 3*x^2 + 5*x + 3)/(1 - x^9). - Chai Wah Wu, Jul 17 2016

Edited: name specified, digital root link added, a comment rewritten and moved to formula section. - Wolfdieter Lang, Jan 05 2015

A251780 Digital root of A069778(n-1) = n^3 - n^2 + 1, n >= 1. Repeat(1, 6, 3, 7, 6, 6, 4, 6, 9).

Original entry on oeis.org

1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9

Offset: 1

Peter M. Chema, Dec 08 2014

Periodic with cycle of 9: {1, 6, 3, 7, 6, 6, 4, 6, 9}.

The decimal expansion of 54588823/333333333 = 0.repeat(163766469).

			For a(3) = 3 because 3^3 - 3^2 + 3  = 27 - 9 + 3 = 21 with digit sum 3 which is also the digital root of 21.

Eric Weisstein's World of Mathematics, Digital Root.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A069778, A251754, A010888, A056992, A073636.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 6, 3, 7, 6, 6, 4, 6, 9},108] (* Ray Chandler, Jul 25 2016 *)

DR(n)=s=sumdigits(n);while(s>9,s=sumdigits(s));s
for(n=1,100,print1(DR(abs(n^2-n-n^3)),", ")) \\ Derek Orr, Dec 30 2014

a(n) = digital root of n^3 - n^2 + n.

More terms from Derek Orr, Dec 30 2014

Edited: name changed; formula, comment and example rewritten; digital root link added. - Wolfdieter Lang, Jan 05 2015

cp's OEIS Frontend

A004164 Sum of digits of n^3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A016791 a(n) = (3*n + 2)^3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A353109 Array read by antidiagonals: A(n, k) is the digital root of n*k with n >= 0 and k >= 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A251754 Digital root of A027444(n) = n + n^2 + n^3, n>=1. Repeat(3, 5, 3, 3, 2, 6, 3, 8, 9).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A251780 Digital root of A069778(n-1) = n^3 - n^2 + 1, n >= 1. Repeat(1, 6, 3, 7, 6, 6, 4, 6, 9).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions