A212200 -id:A212200 - cp's OEIS frontend

A212204 Positions in A212200 where successive new numbers (see A212203) appear.

1, 2, 4, 8, 16, 18, 65, 72, 256, 258, 260, 266, 4101, 4132, 4167, 4290, 65536, 65540, 65542, 65544, 65594, 65600, 65658, 65694

Offset: 1

N. J. A. Sloane, May 03 2012

nonn

			A212200 begins 1, 3, 3, 15, 15, 15, 15, 5, 15, 5, 15, 15, 5, 5, 15, 85, ..., containing 1, 3, 15, 5, 85, ..., which gives the beginning of A212203. The numbers in question appear at positions 1, 2, 4, 8, 16, 18, ..., which is A212204.

J. H. Conway and Alex Ryba, Personal communication, May 03 2012 and Jun 10 2012

More terms from Alex Ryba, Jun 10 2012

A212203 Distinct numbers appearing in A212200 in their order of appearance.

Original entry on oeis.org

1, 3, 15, 5, 85, 255, 51, 17, 21845, 65535, 4369, 13107, 1285, 771, 3855, 257, 1431655765, 4294967295, 84215045, 858993459, 50529027, 286331153, 252645135, 16843009

Offset: 1

N. J. A. Sloane, May 03 2012

nonn

This is a permutation of A094358.

			A212200 begins 1, 3, 3, 15, 15, 15, 15, 5, 15, 5, 15, 15, 5, 5, 15, 85, ..., containing 1, 3, 15, 5, 85, ..., which gives the beginning of A212203. The numbers in question appear at positions 1, 2, 4, 8, 16, 18, ..., which is A212204.

J. H. Conway and Alex Ryba, Personal communication, May 03 2012 and Jun 10 2012

Cf. A212200, A212204, A212201, A212203. A094358.

More terms from Alex Ryba, Jun 10 2012

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

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

A213456 a(n) = smallest m such that the multiplicative order of m in nim-multiplication, A212200(m), is A094358(n).

Original entry on oeis.org

1, 2, 8, 4, 72, 65, 16, 18, 4290, 2416912633124328272, 4132, 4101, 1183088714649826879

Offset: 1

N. J. A. Sloane, Jun 12 2012

nonn

J. H. Conway and Alex Ryba, Personal communication, May 03 2012 and Jun 10 2012

Cf. A212200-A212204, A094358.

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

Original entry on oeis.org

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.

PARI
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See Links section.
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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)

A006042 The nim-square of n.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 5, 4, 13, 12, 14, 15, 11, 10, 8, 9, 24, 25, 27, 26, 30, 31, 29, 28, 21, 20, 22, 23, 19, 18, 16, 17, 52, 53, 55, 54, 50, 51, 49, 48, 57, 56, 58, 59, 63, 62, 60, 61, 44, 45, 47, 46, 42, 43, 41, 40, 33, 32, 34, 35, 39, 38, 36, 37, 103, 102, 100, 101, 97, 96, 98, 99

Offset: 0

N. J. A. Sloane

This is a permutation of the natural numbers; A160679 is the inverse permutation. - Jianing Song, Aug 10 2022

J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, Table of n, a(n) for n = 0..1000
G. P. Michon, Discussion of Conway's On2 [From _John W. Layman_, Nov 05 2010]
Index entries for sequences related to Nim-multiplication
Index entries for sequences that are permutations of the natural numbers

Diagonal of A051775. Without 0, diagonal of A051776.
Column 2 of array in A335162.
Other nim k-th powers: A051917 (k=-1), A160679 (k=1/2), A335170 (k=3), A335535 (k=4), A335171 (k=5), A335172 (k=6), A335173 (k=7), A335536 (k=8).
Cf. A212200, A006017.

read("transforms") ;
# insert source of nimprodP2() and A051775() from the b-file at A051776 here...
A006042 := proc(n)
     A051775(n,n) ;
end proc:
L := [seq( A006042(n),n=1..1000) ]; # R. J. Mathar, May 28 2011

a(n) = A051775(n,n).
From Jianing Song, Aug 10 2022: (Start)
If n = Sum_j 2^e(j), then a(n) is the XOR of A006017(e(j))'s. Proof: let N+ = XOR and N* denote the nim addition and the nim multiplication, then n N* n = (Sum_j 2^e(j)) N* (Sum_j 2^e(j)) = (Nim-sum_j 2^e(j)) N* (Nim-sum_j 2^e(j)) = (Nim-sum_j (2^e(j) N* 2^e(j))) N+ (Nim-sum_{iFor example, for n = 11 = 2^0 + 2^1 + 2^3, a(11) = A006017(0) XOR A006017(1) XOR A006017(3) = 1 XOR 3 XOR 13 = 15.
More generally, if n = Sum_j 2^e(j), k is a power of 2, then the nim k-th power of n is the XOR of (nim k-th power of 2^e(j))'s. (End)

a(1)-a(49) confirmed, a(50)-a(71) added by John W. Layman, Nov 05 2010

a(0) prepended by Jianing Song, Aug 10 2022

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)

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A212204 Positions in A212200 where successive new numbers (see A212203) appear.

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A212203 Distinct numbers appearing in A212200 in their order of appearance.

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A212201 Records in A212200.

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A212202 Where records in A212200 occur.

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A213456 a(n) = smallest m such that the multiplicative order of m in nim-multiplication, A212200(m), is A094358(n).

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

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PARI

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A006042 The nim-square of n.

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Maple

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

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