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

Author

Rémy Sigrist, Aug 10 2022

Keywords

• nonn
• tabl
• base

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)).

Examples

			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
		

Links

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

Crossrefs

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

Programs

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

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.