A343042 Array T(n, k), n, k >= 0, read by antidiagonals; lunar multiplication table for the factorial base.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 2, 8, 8, 2, 0, 0, 3, 6, 9, 6, 3, 0, 0, 6, 8, 8, 8, 8, 6, 0, 0, 7, 24, 9, 12, 9, 24, 7, 0, 0, 8, 26, 30, 14, 14, 30, 26, 8, 0, 0, 9, 30, 33, 24, 15, 24, 33, 30, 9, 0, 0, 8, 32, 32, 26, 30, 30, 26, 32, 32, 8, 0

Offset: 0

Author

Rémy Sigrist, Apr 05 2021

Keywords

• nonn
• base
• tabl

Comments

To compute T(n, k):

- write the factorial base representations of n and of k on two lines, right aligned,

- to "multiply" two digits: take the smallest,

- to "add" two digits: take the largest,

- for example, for T(13, 14):

12 -> 2 0 1

14 -> x 2 1 0

-------

0 0 0

1 0 1

+ 2 0 1

-----------

2 1 1 1 0 -> 272 = T(13, 14)

See A343040 for the corresponding addition table.

Examples

			Array T(n, k) begins:
  n\k|  0  1   2   3   4   5    6    7    8    9   10   11   12
  ---+---------------------------------------------------------
    0|  0  0   0   0   0   0    0    0    0    0    0    0    0
    1|  0  1   2   3   2   3    6    7    8    9    8    9    6
    2|  0  2   6   8   6   8   24   26   30   32   30   32   24
    3|  0  3   8   9   8   9   30   33   32   33   32   33   30
    4|  0  2   6   8  12  14   24   26   30   32   36   38   48
    5|  0  3   8   9  14  15   30   33   32   33   38   39   54
    6|  0  6  24  30  24  30  120  126  144  150  144  150  120
    7|  0  7  26  33  26  33  126  127  152  153  152  153  126
    8|  0  8  30  32  30  32  144  152  150  152  150  152  144
    9|  0  9  32  33  32  33  150  153  152  153  152  153  150
   10|  0  8  30  32  36  38  144  152  150  152  156  158  168
   11|  0  9  32  33  38  39  150  153  152  153  158  159  174
   12|  0  6  24  30  48  54  120  126  144  150  168  174  240
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10010
• Rémy Sigrist, PARI program for A343042
• Index entries for sequences related to dismal (or lunar) arithmetic
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A087062, A343040, A343046.

Programs

PARI

See Links section.

Formula

T(n, k) = T(k, n).

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

T(n, 0) = 0.