A021559 -id:A021559 - cp's OEIS frontend

A061105 Smallest number whose sum of digits is n^3.

0, 1, 8, 999, 19999999, 89999999999999, 999999999999999999999999, 199999999999999999999999999999999999999, 899999999999999999999999999999999999999999999999999999999

Offset: 0

Amarnath Murthy, Apr 20 2001

Except for the leading digit all the other digits of a(n), n >= 1, are 9's and the leading digit is 1 or 8. (This is because the digital sum of n^3 is congruent to 0, 1, or 8 mod 9, so the best we can do is use as many 9's as possible, prefixed if necessary by 1 or 8. - N. J. A. Sloane, Jul 19 2018)

			a(4) = 19999999, 1+9+9+9+9+9+9+9 = 64 = 4^3.

Harry J. Smith, Table of n, a(n) for n=0,...,20

Cf. A051885, A061104.

Do[a = {}; While[Apply[Plus, a] + 9 < n^3, a = Append[a, 9]]; If[ Apply[ Plus, a] != n^3, a = Prepend[ a, n^3 - Apply[ Plus, a]] ]; Print[ FromDigits[ a]], {n, 1, 10} ]
dsn3[n_]:=Module[{t=(n^3-{0,1,8})/9},Which[ IntegerQ[t[[1]]],FromDigits[ PadRight[ {},t[[1]],9]],IntegerQ[t[[2]]],FromDigits[ PadRight[ {1}, t[[2]]+1,9]],IntegerQ[t[[3]]],FromDigits[PadRight[{8},t[[3]]+1,9]]]]; Array[dsn3,10,0] (* Harvey P. Dale, Jul 19 2018 *)

a(n) = { ((n%3)^3 + 1)*10^(n^3\9) - 1 } \\ Harry J. Smith, Jul 19 2009

a(n) = A051885(n^3).

a(n) =((n mod 3)^3+1)*10^floor[n^3/9]-1 =(A021559(n+1)+1)*10^A061263(n)-1. - Henry Bottomley, Apr 24 2001

More terms from Robert G. Wilson v, Apr 21 2001

A167176 a(n) = n^3 mod 9.

Original entry on oeis.org

0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8, 0, 1, 8

Offset: 0

Zerinvary Lajos, Oct 29 2009

Essentially a duplicate of A021559. - N. J. A. Sloane, Oct 30 2009
Equivalently: n^(6*m + 3) mod 9. - G. C. Greubel, Jun 04 2016
Decimal expansion of 2/111. - Elmo R. Oliveira, Feb 19 2024

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Table[Mod[n^3,9],{n,0,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2011 *)

a(n)=n^3%9 \\ Charles R Greathouse IV, Apr 06 2016

a(n)=[0,1,8][n%3+1]; \\ Joerg Arndt, Feb 20 2024

[power_mod(n,3,9 ) for n in range(0, 105)]

G.f.: -x*(1+8*x) / ( (x-1)*(1+x+x^2) ). - R. J. Mathar, Aug 24 2016

A060154 Table T(n,k) by antidiagonals of n^k mod k [n,k >= 1].

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Mar 12 2001

			T(5,3) = 5^3 mod 3 = 125 mod 3 = 2.
Rows start:
  0, 1, 1, 1, 1, ...
  0, 0, 2, 0, 2, ...
  0, 1, 0, 1, 3, ...
  0, 0, 1, 0, 4, ...
  0, 1, 2, 1, 0, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Rows include A057427, A015910, A056969.
Columns include A000004, A000035 (several times), A010872, A010874, A010876, A021559 and other periodic sequences.
Diagonals include A000004 and A057427.
Cf. A114448.

T(n, k) = A051129(n, k)-n*A060155(n, k).

A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Feb 14 2006

Alternate description: triangular array a(n, k) = n^k (mod k) read by rows (n > 1, 0 < k < n). This is equivalent because a(n, k) = a(n-k, k). - David Wasserman, Jan 25 2007

			2^6 = 64 and 64 (mod 6) is 4. So a(2,6) = 4.

Cf. A051127, A051128, A094263. Transpose of A060154. First 13 rows are A057427(n-1), A015910, A066601, A066602, A066603, A066604, A066438, A066439, A066440, A056969, A066441, A066442, A116609. First 12 columns are A000004, A000035, A010872, A000035, A010874, A070431, A010876, A000035, A021559(n+1), A008959, A010880, A070435.

a[n_, k_] := Mod[n^k, k]; Table[a[n - k + 1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

More terms from David Wasserman, Jan 25 2007

cp's OEIS Frontend

A061105 Smallest number whose sum of digits is n^3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A167176 a(n) = n^3 mod 9.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

PARI

Sage

Formula

A060154 Table T(n,k) by antidiagonals of n^k mod k [n,k >= 1].

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions