A006042 The nim-square of n.
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
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
- 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
Crossrefs
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
A335170 Nim-cube of n.
0, 1, 1, 1, 14, 13, 8, 10, 14, 10, 13, 8, 14, 8, 10, 13, 152, 145, 133, 141, 189, 182, 167, 173, 203, 199, 212, 217, 224, 238, 248, 247, 152, 141, 145, 133, 224, 247, 238, 248, 189, 173, 182, 167, 203, 217, 199, 212, 152, 133, 141, 145, 203, 212, 217, 199, 224
Offset: 0
Keywords
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..10000
- J. H. Conway, Integral lexicographic codes, Discrete Mathematics 83.2-3 (1990): 219-235. See Table 3.
- Rémy Sigrist, PARI program for A335170
Crossrefs
A column of the array in A335162.
Programs
-
PARI
\\ See Links section.
Extensions
More terms from Rémy Sigrist, Jun 12 2020
A335172 Nim sixth-power of n.
0, 1, 1, 1, 8, 10, 13, 14, 8, 14, 10, 13, 8, 13, 14, 10, 203, 199, 217, 212, 248, 247, 238, 224, 182, 189, 167, 173, 141, 133, 152, 145, 203, 212, 199, 217, 141, 145, 133, 152, 248, 224, 247, 238, 182, 173, 189, 167, 203, 217, 212, 199, 182, 167, 173, 189, 141
Offset: 0
Keywords
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..10000
- J. H. Conway, Integral lexicographic codes, Discrete Mathematics 83.2-3 (1990): 219-235. See Table 3.
Crossrefs
A column of the array in A335162.
Extensions
More terms from Rémy Sigrist, Jun 12 2020
A335171 Nim fifth-power of n.
0, 1, 3, 2, 2, 2, 3, 3, 1, 2, 1, 2, 3, 1, 1, 3, 72, 76, 65, 69, 88, 93, 83, 86, 106, 108, 96, 102, 122, 125, 114, 117, 196, 207, 200, 195, 229, 239, 235, 225, 244, 253, 251, 242, 213, 221, 216, 208, 140, 130, 138, 132, 191, 176, 187, 180, 159, 147, 154, 150
Offset: 0
Keywords
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..10000
- J. H. Conway, Integral lexicographic codes, Discrete Mathematics 83.2-3 (1990): 219-235. See Table 3.
Crossrefs
A column of the array in A335162.
Extensions
More terms from Rémy Sigrist, Jun 12 2020
A335173 Nim seventh-power of n.
0, 1, 2, 3, 11, 9, 15, 12, 13, 4, 14, 5, 6, 10, 8, 7, 118, 113, 32, 34, 105, 111, 40, 42, 78, 74, 55, 52, 58, 57, 95, 90, 155, 51, 146, 48, 31, 175, 30, 165, 190, 63, 181, 60, 135, 24, 143, 25, 237, 16, 17, 227, 201, 46, 44, 197, 37, 250, 245, 39, 215, 20, 21
Offset: 0
Keywords
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..10000
- J. H. Conway, Integral lexicographic codes, Discrete Mathematics 83.2-3 (1990): 219-235. See Table 3.
Crossrefs
A column of the array in A335162.
Extensions
More terms from Rémy Sigrist, Jun 12 2020
A335535 Nim fourth-power of n.
0, 1, 2, 3, 5, 4, 7, 6, 10, 11, 8, 9, 15, 14, 13, 12, 21, 20, 23, 22, 16, 17, 18, 19, 31, 30, 29, 28, 26, 27, 24, 25, 42, 43, 40, 41, 47, 46, 45, 44, 32, 33, 34, 35, 37, 36, 39, 38, 63, 62, 61, 60, 58, 59, 56, 57, 53, 52, 55, 54, 48, 49, 50, 51, 87, 86, 85, 84
Offset: 0
Keywords
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..10000
- J. H. Conway, Integral lexicographic codes, Discrete Mathematics 83.2-3 (1990): 219-235. See Table 3.
Crossrefs
A column of the array in A335162.
A335536 Nim eighth-power of n.
0, 1, 3, 2, 7, 6, 4, 5, 14, 15, 13, 12, 9, 8, 10, 11, 31, 30, 28, 29, 24, 25, 27, 26, 17, 16, 18, 19, 22, 23, 21, 20, 58, 59, 57, 56, 61, 60, 62, 63, 52, 53, 55, 54, 51, 50, 48, 49, 37, 36, 38, 39, 34, 35, 33, 32, 43, 42, 40, 41, 44, 45, 47, 46, 123, 122, 120
Offset: 0
Keywords
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..10000
- J. H. Conway, Integral lexicographic codes, Discrete Mathematics 83.2-3 (1990): 219-235. See Table 3.
Crossrefs
A column of the array in A335162.
A059971 n^n using Nim multiplication.
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
Keywords
Links
- 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
Programs
-
PARI
See Links section.
Formula
a(n) = A335162(n, n). - Rémy Sigrist, Jun 12 2020
Extensions
a(0) = 1 prepended by Rémy Sigrist, Jun 12 2020
A160679 Square root of n under Nim (or Conway) multiplication.
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
Comments
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).
Examples
a(2) = 3 because TIM(3,3) = 2 More generally, a(x)=y because A006042(y)=x.
Links
- Paul Tek, Table of n, a(n) for n = 0..576
- G. P. Michon, Nim-multiplication in Conway's algebraically complete field On2
- Index entries for sequences that are permutations of the natural numbers
- Index entries for sequences related to Nim-multiplication
- Index entries for sequences that are permutations of the natural numbers
Formula
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).
A333862 a(n) is the greatest nim-power of n.
1, 1, 3, 3, 15, 15, 15, 15, 14, 15, 14, 15, 15, 14, 14, 15, 252, 252, 255, 255, 252, 252, 255, 255, 252, 252, 255, 255, 255, 255, 252, 252, 255, 255, 255, 255, 252, 255, 252, 255, 255, 255, 255, 255, 255, 252, 255, 252, 255, 252, 252, 255, 255, 255, 255, 255
Offset: 0
Keywords
Comments
For any n >= 0, a(n) is the greatest term in n-th row of A335162.
Examples
For n = 10: - the first nim-powers of 10 are: 1, 10, 14, 13, 8, 1, ... - so a(10) = 14.
Links
Crossrefs
Cf. A335162.
Programs
-
C
See Links section.
Comments