A341288 Square array T(n, k), read by antidiagonals, n, k >= 0; T(n, k) = XOR_{u in B(n), v in B(k)} 2^(u XOR v) where XOR denotes the bitwise XOR operator and B(n) gives the exponents in expression for n as a sum of powers of 2.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 1, 3, 0, 0, 4, 3, 3, 4, 0, 0, 5, 8, 0, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 9, 15, 1, 15, 9, 7, 0, 0, 8, 11, 15, 5, 5, 15, 11, 8, 0, 0, 9, 4, 12, 9, 0, 9, 12, 4, 9, 0, 0, 10, 6, 12, 13, 15, 15, 13, 12, 6, 10, 0

Offset: 0

Author

Rémy Sigrist, Feb 08 2021

Keywords

• nonn
• tabl
• look
• base

Comments

For any x >= 0, the function n -> T(n, 2^x) is a self-inverse permutation of the nonnegative integers.

The set of nonnegative integers equipped with T forms a commutative monoid; its invertible elements are the odious numbers (A000069).

Hence A000069 equipped with T forms a group.

Examples

			Array T(n, k) begins:
  n\k|  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+---------------------------------------------------------------
    0|  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
    1|  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    2|  0   2   1   3   8  10   9  11   4   6   5   7  12  14  13  15 -> A057300
    3|  0   3   3   0  12  15  15  12  12  15  15  12   0   3   3   0
    4|  0   4   8  12   1   5   9  13   2   6  10  14   3   7  11  15 -> A126006
    5|  0   5  10  15   5   0  15  10  10  15   0   5  15  10   5   0
    6|  0   6   9  15   9  15   0   6   6   0  15   9  15   9   6   0
    7|  0   7  11  12  13  10   6   1  14   9   5   2   3   4   8  15
    8|  0   8   4  12   2  10   6  14   1   9   5  13   3  11   7  15
    9|  0   9   6  15   6  15   0   9   9   0  15   6  15   6   9   0
   10|  0  10   5  15  10   0  15   5   5  15   0  10  15   5  10   0
   11|  0  11   7  12  14   5   9   2  13   6  10   1   3   8   4  15
   12|  0  12  12   0   3  15  15   3   3  15  15   3   0  12  12   0
   13|  0  13  14   3   7  10   9   4  11   6   5   8  12   1   2  15
   14|  0  14  13   3  11   5   6   8   7   9  10   4  12   2   1  15
   15|  0  15  15   0  15   0   0  15  15   0   0  15   0  15  15   0
                                                                     \
                                                                      v
                                                                    A010060
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10010 (rows 0..140)
• Rémy Sigrist, Colored representation of the table for n, k < 2^10
• Rémy Sigrist, Colored representation of the table over the first 128 odious numbers
• Rémy Sigrist, Colored representation of the table over the first 1024 evil numbers (white pixels correspond to 0's)

Crossrefs

Cf. A000069, A010060, A057300, A126006, A133457.

Programs

PARI

B(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n -= 2^(b[k] = valuation(n, 2))); b }
T(n,k) = { my (nn=B(n), kk=B(k), v=0); for (i=1, #nn, for (j=1, #kk, v=bitxor(v, 2^bitxor(nn[i], kk[j])))); v }

Formula

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

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

T(n, 0) = 0 (0 is an absorbing element for T).

T(n, 1) = n (1 is the neutral element for T).

T(n, 2) = A057300(n).

T(n, 4) = A126006(n).

T(n, n) = A010060(n).

A010060(T(n, k)) = A010060(n) * A010060(k).