A067138 OR-numbral multiplication table, read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 7, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 12, 15, 16, 15, 12, 7, 0, 0, 8, 14, 14, 20, 20, 14, 14, 8, 0, 0, 9, 16, 15, 24, 21, 24, 15, 16, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 20, 27, 32, 31, 28

Offset: 0

Author

Jens Voß, Jan 02 2002

Keywords

• nonn
• tabl
• look

Comments

See A048888 for the definition of OR-numbral arithmetic

Examples

			The top left 0..16 x 0..16 corner of the array:
  0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,
  0,  1,  2,  3,  4,  5,  6,   7,   8,   9,  10,  11,  12,  13,  14,  15,
  0,  2,  4,  6,  8, 10, 12,  14,  16,  18,  20,  22,  24,  26,  28,  30,
  0,  3,  6,  7, 12, 15, 14,  15,  24,  27,  30,  31,  28,  31,  30,  31,
  0,  4,  8, 12, 16, 20, 24,  28,  32,  36,  40,  44,  48,  52,  56,  60,
  0,  5, 10, 15, 20, 21, 30,  31,  40,  45,  42,  47,  60,  61,  62,  63,
  0,  6, 12, 14, 24, 30, 28,  30,  48,  54,  60,  62,  56,  62,  60,  62,
  0,  7, 14, 15, 28, 31, 30,  31,  56,  63,  62,  63,  60,  63,  62,  63,
  0,  8, 16, 24, 32, 40, 48,  56,  64,  72,  80,  88,  96, 104, 112, 120,
  0,  9, 18, 27, 36, 45, 54,  63,  72,  73,  90,  91, 108, 109, 126, 127,
  0, 10, 20, 30, 40, 42, 60,  62,  80,  90,  84,  94, 120, 122, 124, 126,
  0, 11, 22, 31, 44, 47, 62,  63,  88,  91,  94,  95, 124, 127, 126, 127,
  0, 12, 24, 28, 48, 60, 56,  60,  96, 108, 120, 124, 112, 124, 120, 124,
  0, 13, 26, 31, 52, 61, 62,  63, 104, 109, 122, 127, 124, 125, 126, 127,
  0, 14, 28, 30, 56, 62, 60,  62, 112, 126, 124, 126, 120, 126, 124, 126,
  0, 15, 30, 31, 60, 63, 62,  63, 120, 127, 126, 127, 124, 127, 126, 127,
  0, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240,
.
Multiplying 3 ("11" in binary) with itself in this system means taking bitwise-or of "11" with itself, when shifted one bit-position left:
       11
      110
  -------
OR:   111 = 7 in decimal = A(3,3).
.
Multiplying 10 (= "1010" in binary) and 11 (= "1011" in binary) in this system means taking bitwise-or of binary number 1011 when shifted once left with the same binary number when shifted three bit-positions left:
      10110
    1011000
    -------
OR: 1011110 = 94 in decimal = A(10,11) = A(11,10).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10439; the first 144 antidiagonals of square array

Crossrefs

Cf. A003986, A067139, A048888, A007059, A067398 (main diagonal).

Cf. also A004247, A048720 for analogous multiplication tables.

Programs

PARI

t(n, k) = {res = 0; for (i=0, length(binary(n))-1, if (bittest(n, i), res = bitor(res, shift(k, i)));); return (res);} \\ Michel Marcus, Apr 14 2013

Formula

From Rémy Sigrist, Mar 17 2021: (Start)

T(n, 0) = 0.

T(n, 1) = n.

T(n, 2^k) = n*2^k for any k >= 0.

T(n, n) = A067398(n).

(End)

For all n, k: A048720(n,k) <= A(n,k) <= A004247(n,k). - Antti Karttunen, Mar 17 2021

Extensions

Example-section rewritten by Antti Karttunen, Mar 17 2021