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

0,4

Comments

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

Each row is a permutation of the nonnegative integers.

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

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).

Example

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

Prog

(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)) }

Crossrefs

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

Sequence in context: A307297 A307301 A307300 * A354470 A141692 A261097

Adjacent sequences: A354435 A354436 A354437 * A354439 A354440 A354441

Keyword

nonn,tabl,base

Author

Rémy Sigrist, May 28 2022

Status

approved