A223542 -id:A223542 - cp's OEIS frontend

A223541 Array T(m,n) = nim-product(2^m,2^n) (m>=0, n>=0) read by antidiagonals.

1, 2, 2, 4, 3, 4, 8, 8, 8, 8, 16, 12, 6, 12, 16, 32, 32, 11, 11, 32, 32, 64, 48, 64, 13, 64, 48, 64, 128, 128, 128, 128, 128, 128, 128, 128, 256, 192, 96, 192, 24, 192, 96, 192, 256, 512, 512, 176, 176, 44, 44, 176, 176, 512, 512, 1024, 768

Offset: 0

Tilman Piesk, Mar 21 2013

Nimber multiplication is commutative, so this array is symmetric, and can be represented in a more compact way by the rows of the lower triangle (A223540).
The diagonal is A006017 (nim-squares of powers of 2).
The elements of this array are listed in A223543. In the key-matrix A223542 the entries of this array (which become very large) are replaced by the corresponding index numbers of A223543. (Surprisingly, the key-matrix seems to be interesting on its own.)
The number of different entries per antidiagonal is probably A002487. That would mean that there are exactly A002487(n+1) numbers that can be expressed as a nim-product(2^a,2^b) with a+b=n. - Tilman Piesk, Mar 27 2013

			T(1,7) = T(3,5) = 192, the result of the nim-multiplications 2^1*2^7 and 2^3*2^5.
The array begins:
1,2,4,8,16,32,64,128,256,...
2,3,8,12,32,48,128,192,512,...
4,8,6,11,64,128,96,176,1024,...
8,12,11,13,128,192,176,208,2048,...
16,32,64,128,24,44,75,141,4096,...
32,48,128,192,44,52,141,198,8192,...
64,128,96,176,75,141,103,185,16384,...
128,192,176,208,141,198,185,222,32768,...
256,512,1024,2048,4096,8192,16384,32768,384,...
...

J. H. Conway, "Integral lexicographic codes." Discrete Mathematics 83.2-3 (1990): 219-235. See Table 4.

Tilman Piesk, First 128 rows of the matrix, flattened
Tilman Piesk, Elements of dual matrix (256 SVGs)
Tilman Piesk, Walsh permutation; nimber multiplication (Wikiversity)
Tilman Piesk, Class bin and function nimprod (MATLAB code)

Cf. A051775, A223540, A006017 (main diagonal), A223543, A223542, A000079, A002487.

For rows 1,2,3,4, see A134683, A335159, A335160, A335161.

T(m,n) = A051775(A000079(m),A000079(n)).

T(m,n) = A223543(A223542(m,n)).

Edited by N. J. A. Sloane, Jun 08 2020

A223543 Nim-products of powers of 2, list of entries in matrix A223541.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 12, 13, 16, 24, 32, 44, 48, 52, 64, 75, 96, 103, 128, 141, 176, 185, 192, 198, 208, 222, 256, 384, 512, 704, 768, 832, 1024, 1200, 1536, 1648, 2048, 2256, 2816, 2960, 3072, 3168, 3328, 3552, 4096, 4237, 6144, 6237

Offset: 0

Tilman Piesk, Mar 21 2013

nonn

List of entries in the nim-multiplication table of powers of 2 (A223541).

First 3^n entries are the distinct entries of multiplication table of size 2^n.

			a(23) = 192, which is the result of the nim-multiplications 2*128 and 8*32.

Tilman Piesk, Table of n, a(n) for n = 0..6560
Tilman Piesk, Walsh permutation; nimber multiplication (Wikiversity)
Tilman Piesk, Detailed list, including the entries' positions in the matrix

Cf. A051775 (nim-multiplication table).
Cf. A223541 (nim-multiplication table of powers of 2).
Cf. A006017 (nim-squares of powers of 2).
Cf. A006046 (sum of first n rows in Sierpinski's triangle).
Cf. A000079 (powers of 2).
Cf. A000244 (powers of 3).

A223541(m,n) = a( A223542(m,n) ).

a(0,2,4,8,10,14,18,26...) = a( A006046(1,2,3...) - 1 ) = A006017.

cp's OEIS Frontend

A223541 Array T(m,n) = nim-product(2^m,2^n) (m>=0, n>=0) read by antidiagonals.

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