cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A223541 Array T(m,n) = nim-product(2^m,2^n) (m>=0, n>=0) read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 8, 8, 8, 8, 16, 12, 6, 12, 16, 32, 32, 11, 11, 32, 32, 64, 48, 64, 13, 64, 48, 64, 128, 128, 128, 128, 128, 128, 128, 128, 256, 192, 96, 192, 24, 192, 96, 192, 256, 512, 512, 176, 176, 44, 44, 176, 176, 512, 512, 1024, 768
Offset: 0

Views

Author

Tilman Piesk, Mar 21 2013

Keywords

Comments

Nimber multiplication is commutative, so this array is symmetric, and can be represented in a more compact way by the rows of the lower triangle (A223540).
The diagonal is A006017 (nim-squares of powers of 2).
The elements of this array are listed in A223543. In the key-matrix A223542 the entries of this array (which become very large) are replaced by the corresponding index numbers of A223543. (Surprisingly, the key-matrix seems to be interesting on its own.)
The number of different entries per antidiagonal is probably A002487. That would mean that there are exactly A002487(n+1) numbers that can be expressed as a nim-product(2^a,2^b) with a+b=n. - Tilman Piesk, Mar 27 2013

Examples

			T(1,7) = T(3,5) = 192, the result of the nim-multiplications 2^1*2^7 and 2^3*2^5.
The array begins:
1,2,4,8,16,32,64,128,256,...
2,3,8,12,32,48,128,192,512,...
4,8,6,11,64,128,96,176,1024,...
8,12,11,13,128,192,176,208,2048,...
16,32,64,128,24,44,75,141,4096,...
32,48,128,192,44,52,141,198,8192,...
64,128,96,176,75,141,103,185,16384,...
128,192,176,208,141,198,185,222,32768,...
256,512,1024,2048,4096,8192,16384,32768,384,...
...

References

  • J. H. Conway, "Integral lexicographic codes." Discrete Mathematics 83.2-3 (1990): 219-235. See Table 4.

Links

Crossrefs

Cf. A051775, A223540, A006017 (main diagonal), A223543, A223542, A000079, A002487.
For rows 1,2,3,4, see A134683, A335159, A335160, A335161.

Formula

T(m,n) = A051775(A000079(m),A000079(n)).
T(m,n) = A223543(A223542(m,n)).

Extensions

Edited by N. J. A. Sloane, Jun 08 2020

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

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

Views

Author

Tilman Piesk, Mar 21 2013

Keywords

Comments

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.

Examples

			a(23) = 192, which is the result of the nim-multiplications 2*128 and 8*32.

Links

Crossrefs

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).

Formula

A223541(m,n) = a( A223542(m,n) ).
a(0,2,4,8,10,14,18,26...) = a( A006046(1,2,3...) - 1 ) = A006017.

A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

Original entry on oeis.org

1, 3, 6, 15, 24, 60, 105, 255, 384, 960, 1632, 1680, 4080, 15555, 27030, 65535, 98304, 245760, 417792, 430080, 1044480, 1582080, 3947520, 3982080, 6908160, 6919680, 16776960, 106991625, 267448335, 1019462460, 1771476585, 4294967295
Offset: 0

Views

Author

Tilman Piesk, Aug 01 2013

Keywords

Comments

A subsequence of A227723, showing all the big equivalence classes that contain Boolean functions related to subgroups of nimber addition (A190939).
Forms a triangle with row lengths A034343 = 1, 1, 2, 4, 8, 16, 36, 80...:
1,
3,
6, 15,
24, 60, 105, 255,
384, 960, 1632, 1680, 4080, 15555, 27030, 65535...
The left column a( 1,2,4,8,16,32,68,148... ) = a( A076766 ) = 3 ,6, 24, 384, 98304... is probably A001146 * 3/2, which is also A006017( A000079 ).
The first A076766(n) entries correspond to the first A006116(n) entries of A190939. (The first 148 here, for n = 7, correspond to the first 29212 there.) The entries of A190939 can be generated from this sequence.
Among the first A076766(n) entries are A076831(n;0...n) with weight 2^0...2^n. (Among the first 148 are 1, 7, 23, 43, 43, 23, 7, 1 with weights 1, 2, 4, 8, 16, 32, 64, 128.)
a(n) appears to be divisible by 3 for n>0, and the odd part of a(n) is almost always squarefree. - Ralf Stephan, Aug 02 2013

Links

Crossrefs

Subsequence of A227723 (all becs). All entries are also in A227963 (all sona-secs). Neither shares the property of divisibility by 3.
A190939, A034343, A076766, A076831.
The prime factors contain many prime factors of Fermat numbers (A023394).

Formula

a( A076766 - 1 ) = A001146 - 1 = A051179.
a( A076766 ) = A001146 * 3/2 (probably).

A060147 Nim-binomial transform of the Nim-squares sequence {0,1,3,2,6,7,5,4,13,12,14,...}.

Original entry on oeis.org

0, 1, 3, 0, 6, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 52, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 103, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

John W. Layman, Mar 06 2001

Keywords

Comments

The Nim-binomial transform of the Nim-squares consists of the Nim-squares of the terms of the Nim-binomial transform of the integers (given in A048298).
Multiplicative with a(2^e) = A006017(e), a(p^e) = 0 otherwise. - David W. Wilson, Jun 12 2005

Crossrefs

See A048298.

Formula

a(n) = n X n, where Nim-multiplication is used, if n=2^k, else a(n)=0.
Showing 1-5 of 5 results.

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