A006042 -id:A006042 - cp's OEIS frontend

A335162 Array read by upward antidiagonals: T(n,k) (n >= 0, k >= 0) = nim k-th power of n.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 2, 1, 1, 0, 1, 5, 6, 1, 2, 1, 0, 1, 6, 7, 14, 3, 3, 1, 0, 1, 7, 5, 13, 5, 2, 1, 1, 0, 1, 8, 4, 8, 4, 2, 1, 2, 1, 0, 1, 9, 13, 10, 7, 2, 8, 3, 3, 1, 0, 1, 10, 12, 14, 6, 3, 10, 11, 2, 1, 1, 0, 1, 11, 14, 10, 10, 3, 13, 9, 7, 1, 2, 1, 0, 1, 12, 15, 13, 11, 1, 14, 15, 6, 10, 3, 3, 1, 0

Offset: 0

N. J. A. Sloane, Jun 08 2020

Although the nim-addition table (A003987) and nim-multiplication table (A051775) can be found in Conway's "On Numbers and Games", and in the Berlekamp-Conway-Guy "Winning Ways", this exponentiation-table seems to have been omitted.

The n-th row is A212200(n)-periodic. - Rémy Sigrist, Jun 12 2020

			The array begins:
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0, ...,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1, ...,
  1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1,  2, 3, ...,
  1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1,  3, 2, ...,
  1, 4, 6,14, 5, 2, 8,11, 7,10, 3,12,13, 9,15, 1,  4, 6, ...,
  1, 5, 7,13, 4, 2,10, 9, 6, 8, 3,15,14,11,12, 1,  5, 7, ...,
  1, 6, 5, 8, 7, 3,13,15, 4,14, 2,11,10,12, 9, 1,  6, 5, ...,
  1, 7, 4,10, 6, 3,14,12, 5,13, 2, 9, 8,15,11, 1,  7, 4, ...,
  1, 8,13,14,10, 1, 8,13,14,10, 1, 8,13,14,10, 1,  8,13, ...,
  1, 9,12,10,11, 2,14, 4,15,13, 3, 7, 8, 5, 6, 1,  9,12, ...,
  1,10,14,13, 8, 1,10,14,13, 8, 1,10,14,13, 8, 1, 10,14, ...,
  1,11,15, 8, 9, 2,13, 5,12,14, 3, 6,10, 4, 7, 1, 11,15, ...,
  1,12,11,14,15, 3, 8, 6, 9,10, 2, 4,13, 7, 5, 1, 12,11, ...,
  1,13,10, 8,14, 1,13,10, 8,14, 1,13,10, 8,14, 1, 13,10, ...,
  1,14, 8,10,13, 1,14, 8,10,13, 1,14, 8,10,13, 1, 14, 8, ...,
  1,15, 9,13,12, 3,10, 7,11, 8, 2, 5,14, 6, 4, 1, 15, 9, ...
  ...
The initial antidiagonals are:
  [1]
  [1,  0]
  [1,  1,  0]
  [1,  2,  1,  0]
  [1,  3,  3,  1,  0]
  [1,  4,  2,  1,  1,  0]
  [1,  5,  6,  1,  2,  1,  0]
  [1,  6,  7, 14,  3,  3,  1,  0]
  [1,  7,  5, 13,  5,  2,  1,  1,  0]
  [1,  8,  4,  8,  4,  2,  1,  2,  1,  0]
  [1,  9, 13, 10,  7,  2,  8,  3,  3,  1,  0]
  [1, 10, 12, 14,  6,  3, 10, 11,  2,  1,  1,  0]
  [1, 11, 14, 10, 10,  3, 13,  9,  7,  1,  2,  1,  0]
  [1, 12, 15, 13, 11,  1, 14, 15,  6, 10,  3,  3,  1,  0]
  ...

Rémy Sigrist, Table of n, a(n) for n = 0..20300
J. H. Conway, Integral lexicographic codes, Discrete Mathematics 83.2-3 (1990): 219-235. See Table 3.
Rémy Sigrist, PARI program for A335162
Index entries for sequences related to Nim-multiplication

Cf. A003987, A051775, A059971, A212200, A223541.
Rows: for nim-powers of 4 through 10 see A335163-A335169.
Columns: for nim-squares, cubes, fourth, fifth, sixth, seventh and eighth powers see A006042, A335170, A335535, A335171, A335172, A335173 and A335536.

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From Rémy Sigrist, Jun 12 2020: (Start)
T(n, A212200(n)) = 1 for any n > 0.
T(n, n) = A059971(n).
(End)

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

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Corrected by Gerald McGarvey, Nov 12 2005

A059971 n^n using Nim multiplication.

Original entry on oeis.org

1, 1, 3, 1, 5, 2, 13, 12, 14, 13, 1, 6, 13, 8, 13, 1, 17, 8, 158, 155, 72, 170, 198, 48, 145, 208, 165, 25, 55, 205, 171, 206, 55, 158, 6, 140, 151, 53, 113, 252, 191, 254, 228, 26, 116, 130, 146, 243, 145, 118, 72, 14, 75, 115, 20, 69, 60, 177, 121, 99, 171, 169, 170

Offset: 0

John W. Layman, Mar 05 2001

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A059971
Index entries for sequences related to Nim-multiplication

Cf. A058734, A006042, A051917, A335162.

PARI
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a(n) = A335162(n, n). - Rémy Sigrist, Jun 12 2020

a(0) = 1 prepended by Rémy Sigrist, Jun 12 2020

A160679 Square root of n under Nim (or Conway) multiplication.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 5, 14, 15, 13, 12, 9, 8, 10, 11, 30, 31, 29, 28, 25, 24, 26, 27, 16, 17, 19, 18, 23, 22, 20, 21, 57, 56, 58, 59, 62, 63, 61, 60, 55, 54, 52, 53, 48, 49, 51, 50, 39, 38, 36, 37, 32, 33, 35, 34, 41, 40, 42, 43, 46, 47, 45, 44, 124, 125, 127, 126, 123, 122, 120

Offset: 0

Gerard P. Michon, Jun 25 2009

Because Conway's field On2 (endowed with Nim-multiplication and [bitwise] Nim-addition) has characteristic 2, the Nim-square function (A006042) is an injective field homomorphism (i.e., the square of a sum is the sum of the squares). Thus the square function is a bijection within any finite additive subgroup of On2 (which is a fancy way to say that an integer and its Nim-square have the same bit length). Therefore the Nim square-root function is also a field homomorphism (the square-root of a Nim-sum is the Nim-sum of the square roots) which can be defined as the inverse permutation of A006042 (as such, it preserves bit-length too).

			a(2) = 3 because TIM(3,3) = 2
More generally, a(x)=y because A006042(y)=x.

Cf. A006042 (Nim-squares). A051917 (Nim-reciprocals), A335162, A212200.

Letting NIM (= XOR) TIM and RIM denote respectively the sum, product and square root in Conway's Nim-field On2, we see that the bit-length of NIM(x,TIM(x,x)) is less than that of the positive integer x. This remark turns the following relations into an effective recursive definition of a(n) = RIM(n) which uses the fact that RIM is a field homomorphism in On2:
a(0) = 0
a(n) = NIM(n, a(NIM(n, a(n, TIM(n,n)) )
Note: TIM(n,n) = A006042(n)
From Jianing Song, Aug 10 2022: (Start)
For 0 <= n <= 2^2^k-1, a(n) = A335162(n, 2^(2^k-1)). This is because {0,1,...,2^2^k-1} together with the nim operations makes a field isomorphic to GF(2^2^k).
Also for n > 0, a(n) = A335162(n, (A212200(n)+1)/2). (End)

cp's OEIS Frontend

A335162 Array read by upward antidiagonals: T(n,k) (n >= 0, k >= 0) = nim k-th power of 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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A059971 n^n using Nim multiplication.

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A160679 Square root of n under Nim (or Conway) multiplication.

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