A051775 -id:A051775 - cp's OEIS frontend

A004474 Nim product 9 * n.

0, 9, 14, 7, 15, 6, 1, 8, 5, 12, 11, 2, 10, 3, 4, 13, 144, 153, 158, 151, 159, 150, 145, 152, 149, 156, 155, 146, 154, 147, 148, 157, 224, 233, 238, 231, 239, 230, 225, 232, 229, 236, 235, 226, 234, 227, 228, 237, 112, 121, 126, 119, 127, 118, 113, 120, 117, 124

Offset: 0

N. J. A. Sloane

nonn

J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

R. J. Mathar, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Nim-multiplication

a(n) = A051775(9,n) = A051775(n,9).

More terms from Erich Friedman

A058734 Nim-product n*(n+1).

Original entry on oeis.org

0, 2, 1, 12, 2, 8, 3, 15, 5, 11, 4, 14, 7, 1, 6, 240, 8, 42, 9, 100, 10, 32, 11, 231, 13, 35, 12, 102, 15, 41, 14, 124, 20, 38, 21, 168, 22, 44, 23, 107, 17, 47, 16, 170, 19, 37, 18, 184, 28, 14, 29, 192, 30, 4, 31, 131, 25, 7, 24, 194, 27, 13, 26, 214, 39, 165, 38, 203

Offset: 0

N. J. A. Sloane, Dec 31 2000

J. H. Conway, On Numbers and Games, Academic Press, p. 52.

R. J. Mathar, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Nim-multiplication

A diagonal of A051775. Cf. A006042.

Nim-sum (A003987) of n and A006042(n).

a(n) = A051775(n,n+1).

More terms from John W. Layman, Mar 05 2001

A059970 Nim-factorials: a(1)=1 and, for n>1, a(n)=na(n-1), where denotes Nim multiplication.

Original entry on oeis.org

1, 2, 1, 4, 2, 11, 1, 8, 5, 9, 2, 4, 9, 4, 1, 16, 8, 140, 5, 82, 9, 145, 2, 44, 6, 108, 9, 154, 13, 209, 1, 32, 20, 132, 10, 243, 172, 123, 4, 139, 68, 62, 11, 222, 182, 92, 2, 16, 36, 224, 5, 242, 91, 24, 11, 105, 178, 56, 5, 241, 92, 205, 1, 64, 39, 20, 23, 161, 225, 53

Offset: 1

John W. Layman, Mar 05 2001

nonn

Conjectures:
(1) Nim-Factorial(2^n-1)=1 (verified for n=1,2,3,...,16).
(2) Nim-Factorial(2^n+2^(n-1)-1)=2 (verified for n=1,2,3,...,15).

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, PARI program for A059970

Cf. A058734, A006042, A051917.

A059970 := proc(n)
    option remember;
    if n =1 then
        1;
    else
        A051775(n,procname(n-1)) ;
    end if;
end proc: # R. J. Mathar, Jul 28 2016 based on the program in b051775.txt

PARI
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\\ See Links section.
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Corrected by Gerald McGarvey, Nov 12 2005

A212201 Records in A212200.

Original entry on oeis.org

1, 3, 15, 85, 255, 21845, 65535, 1431655765, 4294967295

Offset: 1

N. J. A. Sloane, May 03 2012

Cf. A212200, A212202, A212203, A212204, A051775.

a(7)-a(9) from Alex Ryba, Jun 10 2012

A223537 Compressed nim-multiplication table read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 4, 3, 8, 8, 8, 12, 12, 16, 12, 5, 4, 192, 32, 32, 10, 10, 64, 64, 64, 48, 128, 15, 160, 128, 240, 128, 128, 192, 192, 240, 240, 80, 80, 256, 192, 80, 64, 17, 80, 96, 160, 20480, 512, 512, 160, 160, 34, 34, 176, 176, 40960, 40960, 1024

Offset: 0

Tilman Piesk, Mar 21 2013

A nim-multiplication table (A051775) of size 2^2^n can be compressed to a matrix of size 2^n, using Walsh permutations. As the nim-multiplication tables are submatrices to the bigger ones, also the compressions are submatrices to the bigger ones, leading to this infinite array.
This array is closely related to the nim-multiplication table powers of 2 (A223541). Both arrays can be seen as different views of the same cubic binary tensor.
The diagonal is A001317 (Sierpinski triangle rows read like binary numbers).
The elements of this array are listed in A223539. In the key-matrix A223538 the entries of this array (which become very large) are replaced by the corresponding index numbers of A223539. (Surprisingly, the key-matrix seems to be interesting on its own.)

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, MATLAB functions cv2cm and cm2cv

Cf. A051775, A223541, A001317, A223539, A223538.

a(m,n) = A223539( A223538(m,n) ).
a(n,n) = A001317(n).
a(1,n) = A134683(n+1).

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.

A004469 Nim product 4 * n.

Original entry on oeis.org

0, 4, 8, 12, 6, 2, 14, 10, 11, 15, 3, 7, 13, 9, 5, 1, 64, 68, 72, 76, 70, 66, 78, 74, 75, 79, 67, 71, 77, 73, 69, 65, 128, 132, 136, 140, 134, 130, 142, 138, 139, 143, 131, 135, 141, 137, 133, 129, 192, 196, 200, 204, 198, 194, 206, 202, 203, 207, 195, 199, 205, 201

Offset: 0

N. J. A. Sloane

nonn

J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

R. J. Mathar, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Nim-multiplication

a(n) = A051775(4,n).

More terms from Erich Friedman

A051911 Triangle T(n,m) = Nim-product of n and m, read by rows, 1<=n<=m.

Original entry on oeis.org

1, 2, 3, 3, 1, 2, 4, 8, 12, 6, 5, 10, 15, 2, 7, 6, 11, 13, 14, 8, 5, 7, 9, 14, 10, 13, 3, 4, 8, 12, 4, 11, 3, 7, 15, 13, 9, 14, 7, 15, 6, 1, 8, 5, 12, 10, 15, 5, 3, 9, 12, 6, 1, 11, 14, 11, 13, 6, 7, 12, 10, 1, 9, 2, 4, 15, 12, 4, 8, 13, 1, 9, 5, 6, 10, 2, 14, 11, 13, 6, 11, 9, 4, 15, 2, 14, 3, 8

Offset: 1

N. J. A. Sloane, Dec 20 1999

A051910 with the first column (the zeros) removed.

			Triangle starts
1;
2,  3;
3,  1,  2;
4,  8, 12,  6;
5, 10, 15,  2,  7;
6, 11, 13, 14,  8, 5;
7,  9, 14, 10, 13, 3,  4;
8, 12,  4, 11,  3, 7, 15, 13;

J. H. Conway, On Numbers and Games, Academic Press, p. 52.

R. J. Mathar, Table of n, a(n) for n = 1..3240
Index entries for sequences related to Nim-multiplication

Cf. A051776, A003987, A051775, A051776, A051910.

More terms (taken from the Conway reference) from Joshua Zucker, May 03 2006

A212202 Where records in A212200 occur.

Original entry on oeis.org

1, 2, 4, 16, 18, 256, 258, 65536, 65540

Offset: 1

N. J. A. Sloane, May 03 2012

Cf. A212200, A212201, A212203, A212204, A051775.

a(7)-a(9) from Alex Ryba, Jun 10 2012

A334290 Array read by upward antidiagonals: T(n,k) (n > 0, k > 0) = nim-division of n by k.

Original entry on oeis.org

1, 2, 3, 3, 1, 2, 4, 2, 3, 15, 5, 12, 1, 5, 12, 6, 15, 8, 10, 4, 9, 7, 13, 10, 1, 8, 14, 11, 8, 14, 11, 14, 13, 7, 13, 10, 9, 4, 9, 4, 1, 15, 6, 15, 6, 10, 7, 12, 11, 9, 6, 7, 5, 11, 8, 11, 5, 14, 2, 5, 1, 12, 3, 13, 12, 7, 12, 6, 15, 13, 6, 8, 10, 9, 14, 4, 9, 5

Offset: 1

Rémy Sigrist, Jun 13 2020

Each row and each column is a permutation of the natural numbers.

			The array begins:
  n\k|   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+------------------------------------------------------------
    1|   1   3   2  15  12   9  11  10   6   8   7   5  14  13   4 --> A051917(n)
    2|   2   1   3   5   4  14  13  15  11  12   9  10   7   6   8
    3|   3   2   1  10   8   7   6   5  13   4  14  15   9  11  12
    4|   4  12   8   1  13  15   7   3  14  11  10   2   5   9   6
    5|   5  15  10  14   1   6  12   9   8   3  13   7  11   4   2
    6|   6  13  11   4   9   1  10  12   5   7   3   8   2  15  14
    7|   7  14   9  11   5   8   1   6   3  15   4  13  12   2  10
    8|   8   4  12   2   6   5   9   1   7  13  15   3  10  14  11
    9|   9   7  14  13  10  12   2  11   1   5   8   6   4   3  15
   10|  10   5  15   7   2  11   4  14  12   1   6   9  13   8   3
   11|  11   6  13   8  14   2  15   4  10   9   1  12   3   5   7
   12|  12   8   4   3  11  10  14   2   9   6   5   1  15   7  13
   13|  13  11   6  12   7   3   5   8  15  14   2   4   1  10   9
   14|  14   9   7   6  15   4   3  13   2  10  12  11   8   1   5
   15|  15  10   5   9   3  13   8   7   4   2  11  14   6  12   1
             |   |   |   |   |
             |   |   |   |   A004474(n)
             |   |   |   A004477(n)
             |   |   A004480(n)
             |   A006015(n)
             A004468(n)

Rémy Sigrist, Table of n, a(n) for n = 1..5050
Rémy Sigrist, PARI program for A334290
Index entries for sequences related to Nim-multiplication

Cf. A004468, A004474, A004477, A004480, A006015, A051775, A051776, A051917, A334291.

PARI
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See Links section.
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T(n, k) = A051776(n, A051917(k)).
T(n, 1) = n.
T(1, n) = A051917(k).
T(n, n) = 1.

cp's OEIS Frontend

A004474 Nim product 9 * n.

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A058734 Nim-product n*(n+1).

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A059970 Nim-factorials: a(1)=1 and, for n>1, a(n)=n*a(n-1), where * denotes Nim multiplication.

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Maple

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A212201 Records in A212200.

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A223537 Compressed nim-multiplication table read by antidiagonals.

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

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A004469 Nim product 4 * n.

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A051911 Triangle T(n,m) = Nim-product of n and m, read by rows, 1<=n<=m.

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Comments

Examples

References

Links

Crossrefs

Extensions

A212202 Where records in A212200 occur.

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Crossrefs

Extensions

A334290 Array read by upward antidiagonals: T(n,k) (n > 0, k > 0) = nim-division of n by k.

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Author

Keywords

A059970 Nim-factorials: a(1)=1 and, for n>1, a(n)=na(n-1), where denotes Nim multiplication.