A341288 -id:A341288 - cp's OEIS frontend

A341458 Unique square array T(n, k) read by antidiagonals, n, k > 0, such that A000069(T(n, k)) = A341288(A000069(n), A000069(k)).

1, 2, 2, 3, 1, 3, 4, 5, 5, 4, 5, 6, 1, 6, 5, 6, 3, 7, 7, 3, 6, 7, 4, 2, 1, 2, 4, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 7, 4, 2, 1, 2, 4, 7, 9, 10, 17, 6, 3, 7, 7, 3, 6, 17, 10, 11, 18, 33, 5, 6, 1, 6, 5, 33, 18, 11, 12, 22, 39, 57, 4, 5, 5, 4, 57, 39, 22, 12

Offset: 1

Rémy Sigrist, Feb 12 2021

The positive integers equipped with T form a group.

Every row (and column) is a self-inverse permutation of the positive integers.

			Array T(n, k) begins:
  n\k|   1   2   3   4   5   6    7    8    9   10   11   12   13   14   15   16
  ---+--------------------------------------------------------------------------
    1|   1   2   3   4   5   6    7    8    9   10   11   12   13   14   15   16
    2|   2   1   5   6   3   4    8    7   17   18   22   21   20   19   23   24
    3|   3   5   1   7   2   8    4    6   33   39   35   37   36   38   34   40
    4|   4   6   7   1   8   2    3    5   57   63   62   60   61   59   58   64
    5|   5   3   2   8   1   7    6    4   65   71   70   68   69   67   66   72
    6|   6   4   8   2   7   1    5    3   89   95   91   93   92   94   90   96
    7|   7   8   4   3   6   5    1    2  105  106  110  109  108  107  111  112
    8|   8   7   6   5   4   3    2    1  113  114  115  116  117  118  119  120
    9|   9  17  33  57  65  89  105  113    1   25   41   49   73   81   97  121
   10|  10  18  39  63  71  95  106  114   25    1   56   48   88   80  121   97
   11|  11  22  35  62  70  91  110  115   41   56    1   32  104  121   80   81
   12|  12  21  37  60  68  93  109  116   49   48   32    1  121  104   88   73
   13|  13  20  36  61  69  92  108  117   73   88  104  121    1   32   48   49
   14|  14  19  38  59  67  94  107  118   81   80  121  104   32    1   56   41
   15|  15  23  34  58  66  90  111  119   97  121   80   88   48   56    1   25
   16|  16  24  40  64  72  96  112  120  121   97   81   73   49   41   25    1

Rémy Sigrist, Table of n, a(n) for n = 1..8385
Rémy Sigrist, Colored representation of the table for 1 <= n, k <= 128
Rémy Sigrist, PARI program for A341458

Cf. A000069, A341288.

See A341487 and A341489 for the second and third rows, respectively.

PARI
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See Links section.
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T(n, k) = T(k, n).
T(n, 1) = n.
T(n, n) = 1.

A370049 Square array A(n, k), n, k >= 0, read by antidiagonals; for any n and k >= 0 with respective binary expansions Sum_{i > 0} b_i2^(i-1) and Sum_{i > 0} c_i2^(i-1), the binary expansion of A(n, k) is Sum_{i > 0} d_i2^(i-1) with d_i = (Sum_{k divides i} b_kc_{i/k}) mod 2 for any i > 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 8, 3, 0, 0, 4, 10, 10, 4, 0, 0, 5, 32, 9, 32, 5, 0, 0, 6, 34, 36, 36, 34, 6, 0, 0, 7, 40, 39, 256, 39, 40, 7, 0, 0, 8, 42, 46, 260, 260, 46, 42, 8, 0, 0, 9, 128, 45, 288, 257, 288, 45, 128, 9, 0, 0, 10, 130, 136, 292, 294, 294, 292, 136, 130, 10, 0

Offset: 0

Rémy Sigrist, Apr 30 2024

The set of nonnegative integers equipped with A form a commutative monoid.

			Array A(n, k) begins:
  n\k | 0   1    2    3     4     5     6     7      8      9     10
  ----+-------------------------------------------------------------
    0 | 0   0    0    0     0     0     0     0      0      0      0
    1 | 0   1    2    3     4     5     6     7      8      9     10
    2 | 0   2    8   10    32    34    40    42    128    130    136
    3 | 0   3   10    9    36    39    46    45    136    139    130
    4 | 0   4   32   36   256   260   288   292   2048   2052   2080
    5 | 0   5   34   39   260   257   294   291   2056   2061   2090
    6 | 0   6   40   46   288   294   264   270   2176   2182   2216
    7 | 0   7   42   45   292   291   270   265   2184   2191   2210
    8 | 0   8  128  136  2048  2056  2176  2184  32768  32776  32896
    9 | 0   9  130  139  2052  2061  2182  2191  32776  32769  32906
   10 | 0  10  136  130  2080  2090  2216  2210  32896  32906  32776

Cf. A000120, A062880, A048720, A341288.

bits(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n-=2^b[k]=valuation(n,2)); return (b); }
A(n, k) = { my (bn = bits(2*n), bk = bits(2*k), v = 0, e); for (i = 1, #bn, for (j = 1, #bk, e = bn[i] * bk[j] - 1; v = bitxor(v, 2^e););); return (v); }

A(n, k) = A(k, n).
A(m, A(n, k)) = A(A(m, n), k).
A(m XOR n, k) = A(m, k) XOR A(n, k) (where XOR denotes the bitwise XOR operator).
A000120(A(n, 2^k)) = A000120(n).
A(n, 0) = 0.
A(n, 1) = n.
A(n, 2) = A062880(n).

cp's OEIS Frontend

A341458 Unique square array T(n, k) read by antidiagonals, n, k > 0, such that A000069(T(n, k)) = A341288(A000069(n), A000069(k)).

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A370049 Square array A(n, k), n, k >= 0, read by antidiagonals; for any n and k >= 0 with respective binary expansions Sum_{i > 0} b_i2^(i-1) and Sum_{i > 0} c_i2^(i-1), the binary expansion of A(n, k) is Sum_{i > 0} d_i2^(i-1) with d_i = (Sum_{k divides i} b_kc_{i/k}) mod 2 for any i > 0.

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cp's OEIS Frontend

A341458 Unique square array T(n, k) read by antidiagonals, n, k > 0, such that A000069(T(n, k)) = A341288(A000069(n), A000069(k)).

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A370049 Square array A(n, k), n, k >= 0, read by antidiagonals; for any n and k >= 0 with respective binary expansions Sum_{i > 0} b_i*2^(i-1) and Sum_{i > 0} c_i*2^(i-1), the binary expansion of A(n, k) is Sum_{i > 0} d_i*2^(i-1) with d_i = (Sum_{k divides i} b_k*c_{i/k}) mod 2 for any i > 0.

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A370049 Square array A(n, k), n, k >= 0, read by antidiagonals; for any n and k >= 0 with respective binary expansions Sum_{i > 0} b_i2^(i-1) and Sum_{i > 0} c_i2^(i-1), the binary expansion of A(n, k) is Sum_{i > 0} d_i2^(i-1) with d_i = (Sum_{k divides i} b_kc_{i/k}) mod 2 for any i > 0.