A354438 Square array A(n, k), n, k >= 0, read by antidiagonals; the factorial base expansion of A(n, k) is obtained by adding componentwise and reducing modulo their radix the digits of the factorial base expansions of n and k.

0, 1, 1, 2, 0, 2, 3, 3, 3, 3, 4, 2, 4, 2, 4, 5, 5, 5, 5, 5, 5, 6, 4, 0, 4, 0, 4, 6, 7, 7, 1, 1, 1, 1, 7, 7, 8, 6, 8, 0, 2, 0, 8, 6, 8, 9, 9, 9, 9, 3, 3, 9, 9, 9, 9, 10, 8, 10, 8, 10, 2, 10, 8, 10, 8, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Author

Rémy Sigrist, May 28 2022

Keywords

• nonn
• tabl
• base

Comments

The nonnegative integers together with A form an abelian group; A225901 gives inverse elements.

Each row is a permutation of the nonnegative integers.

Examples

			Square array A(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+----------------------------------------------------------------
    0|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    1|   1   0   3   2   5   4   7   6   9   8  11  10  13  12  15  14
    2|   2   3   4   5   0   1   8   9  10  11   6   7  14  15  16  17
    3|   3   2   5   4   1   0   9   8  11  10   7   6  15  14  17  16
    4|   4   5   0   1   2   3  10  11   6   7   8   9  16  17  12  13
    5|   5   4   1   0   3   2  11  10   7   6   9   8  17  16  13  12
    6|   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21
    7|   7   6   9   8  11  10  13  12  15  14  17  16  19  18  21  20
    8|   8   9  10  11   6   7  14  15  16  17  12  13  20  21  22  23
    9|   9   8  11  10   7   6  15  14  17  16  13  12  21  20  23  22
   10|  10  11   6   7   8   9  16  17  12  13  14  15  22  23  18  19
   11|  11  10   7   6   9   8  17  16  13  12  15  14  23  22  19  18
   12|  12  13  14  15  16  17  18  19  20  21  22  23   0   1   2   3
   13|  13  12  15  14  17  16  19  18  21  20  23  22   1   0   3   2
   14|  14  15  16  17  12  13  20  21  22  23  18  19   2   3   4   5
   15|  15  14  17  16  13  12  21  20  23  22  19  18   3   2   5   4
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
• Rémy Sigrist, Colored representation of the array A(n, k) for n, k < 7! (the hue is function of A(n, k), black pixels correspond to 0's)
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A003987, A004442, A108731, A225901, A354470 (primorial base analog).

Programs

PARI

A(n,k, s=i->i+1) = { my (v=0, f=1, r); for (i=1, oo, if (n==0 && k==0, return (v), r=s(i); v+=f*((n+k)%r); f*=r; n\=r; k\=r)) }

Formula

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

A(m, A(n, k)) = A(A(m, n), k).

A(n, 0) = n.

A(n, k) = 0 iff k = A225901(n).

A(n, 1) = A004442(n).