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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

Views

Author

N. J. A. Sloane, Jun 08 2020

Keywords

Comments

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

Examples

			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]
  ...

Links

Crossrefs

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.

Programs

  • PARI
    See Links section.

Formula

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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).

Links

Crossrefs

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.

Programs

Formula

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_{i
For 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)

Extensions

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

A356522 Numbers that are nim cubes; numbers in A335170.

Original entry on oeis.org

0, 1, 8, 10, 13, 14, 16, 17, 20, 21, 24, 25, 30, 31, 36, 38, 45, 47, 49, 50, 61, 62, 72, 74, 76, 78, 88, 90, 93, 95, 105, 106, 108, 111, 113, 114, 117, 118, 128, 130, 131, 133, 136, 138, 139, 141, 145, 151, 152, 158, 160, 161, 163, 167, 169, 170, 171, 173, 177, 182, 186
Offset: 1

Views

Author

Jianing Song, Aug 10 2022

Keywords

Comments

Also numbers in A335172, or numbers that are nim (3*2^m)-th powers for each m.
There are (2^2^k - 1)/3 + 1 terms <= 2^2^k - 1 for each 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).

Examples

			8 is a term because (6 N* 6) N* 6 = 5 N* 6 = 8, where N* denotes the nim multiplication.

Links

Crossrefs

Cf. A051175, A335170, A335172. See also A335162 for nim powers.

Programs

Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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