A354470 Square array A(n, k), n, k >= 0, read by antidiagonals; the primorial base expansion of A(n, k) is obtained by adding componentwise and reducing modulo their radix the digits of the primorial 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, Jun 02 2022

Keywords

• nonn
• base
• tabl

Comments

The nonnegative integers together with A form an abelian group; A354469 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  24  25  26  27
   13|  13  12  15  14  17  16  19  18  21  20  23  22  25  24  27  26
   14|  14  15  16  17  12  13  20  21  22  23  18  19  26  27  28  29
   15|  15  14  17  16  13  12  21  20  23  22  19  18  27  26  29  28
		

Links

• Rémy Sigrist, Colored representation of the array A(n, k) for n, k < 2*3*5*7*11 (the hue is function of A(n, k), black pixels correspond to 0's)
• Index entries for sequences related to primorial base

Crossrefs

Cf. A004442, A235168, A354438 (factorial base analog), A354469.

Programs

PARI

A(n,k, s=i->prime(i)) = { 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 = A354469(n).

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