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.

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

0,8

Comments

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

Links

Table of n, a(n) for n=0..77.

Formula

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).

Example

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

Prog

(PARI) 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); }

Crossrefs

Cf. A000120, A062880, A048720, A341288.

Sequence in context: A357144 A351786 A329331 * A355664 A254040 A376725

Adjacent sequences: A370046 A370047 A370048 * A370050 A370051 A370052

Keyword

nonn,base,tabl

Author

Rémy Sigrist, Apr 30 2024

Status

approved