A356517 Square array A(n, k), n >= 2, k >= 0, read by antidiagonals upwards; A(n, k) is the least integer with sum of digits k in base n.

0, 0, 1, 0, 1, 3, 0, 1, 2, 7, 0, 1, 2, 5, 15, 0, 1, 2, 3, 8, 31, 0, 1, 2, 3, 7, 17, 63, 0, 1, 2, 3, 4, 11, 26, 127, 0, 1, 2, 3, 4, 9, 15, 53, 255, 0, 1, 2, 3, 4, 5, 14, 31, 80, 511, 0, 1, 2, 3, 4, 5, 11, 19, 47, 161, 1023, 0, 1, 2, 3, 4, 5, 6, 17, 24, 63, 242, 2047

Offset

2,6

Comments

The expansion of A(n, k) in base n is:

q n-1 ... n-1

<- p times ->

where q = k mod (n-1) and p = floor(k / (n-1)).

Links

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 antidiagonals)

Formula

A(2, k) = 2^k - 1.

A(3, k) = A062318(k+1).

A(4, k) = A180516(k+1).

A(5, k) = A181287(k+1).

A(6, k) = A181288(k+1).

A(7, k) = A181303(k+1).

A(8, k) = A165804(k+1).

A(9, k) = A140576(k+1).

A(10, k) = A051885(k).

A(n, 0) = 0.

A(n, 1) = 1.

A(n, k) = k iff k < n.

A(n, n) = 2*n - 1.

A(n, n+1) = 3*n - 1 for any n > 2.

Example

Array A(n, k) begins:
  n\k|  0  1  2  3   4   5   6    7    8    9    10    11    12
  ---+---------------------------------------------------------
    2|  0  1  3  7  15  31  63  127  255  511  1023  2047  4095
    3|  0  1  2  5   8  17  26   53   80  161   242   485   728
    4|  0  1  2  3   7  11  15   31   47   63   127   191   255
    5|  0  1  2  3   4   9  14   19   24   49    74    99   124
    6|  0  1  2  3   4   5  11   17   23   29    35    71   107
    7|  0  1  2  3   4   5   6   13   20   27    34    41    48
    8|  0  1  2  3   4   5   6    7   15   23    31    39    47
    9|  0  1  2  3   4   5   6    7    8   17    26    35    44
   10|  0  1  2  3   4   5   6    7    8    9    19    29    39
Array A(n, k) begins (with values given in base n):
  n\k|  0  1   2    3     4      5       6        7         8          9
  ---+------------------------------------------------------------------
    2|  0  1  11  111  1111  11111  111111  1111111  11111111  111111111
    3|  0  1   2   12    22    122     222     1222      2222      12222
    4|  0  1   2    3    13     23      33      133       233        333
    5|  0  1   2    3     4     14      24       34        44        144
    6|  0  1   2    3     4      5      15       25        35         45
    7|  0  1   2    3     4      5       6       16        26         36
    8|  0  1   2    3     4      5       6        7        17         27
    9|  0  1   2    3     4      5       6        7         8         18
   10|  0  1   2    3     4      5       6        7         8          9

Prog

(PARI) A(n, k) = { (1+k%(n-1))*n^(k\(n-1))-1 }
(Python) def A(n, k): return (1+(k % (n-1)))*n**(k//(n-1))-1

Crossrefs

Cf. A000225, A051885, A062318, A140576, A165804, A180516, A181287, A181288, A181303, A138530, A240236.

Sequence in context: A014573 A067166 A125209 * A263313 A071818 A014513

Adjacent sequences: A356514 A356515 A356516 * A356518 A356519 A356520

Keyword

nonn,tabl,base

Author

Rémy Sigrist, Aug 10 2022

Status

approved