A190939 -id:A190939 - cp's OEIS frontend

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

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

Tilman Piesk, Aug 01 2013

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

Tilman Piesk, Table of n, a(n) for n = 0..147
Tilman Piesk, a(n) for n = 0..147 and corresponding entries of A190939 in reverse binary. Prime factors and numbers of prime factors (A001222).
Tilman Piesk, Subgroups of nimber addition (Wikiversity)

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).

a( A076766 - 1 ) = A001146 - 1 = A051179.

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

A227963 Small equivalence classes (A227722) of subgroups of nimber addition (A190939).

Original entry on oeis.org

1, 3, 5, 6, 15, 17, 18, 51, 20, 85, 105, 24, 102, 90, 60, 255, 257, 258, 771, 260, 1285, 1545, 264, 1542, 1290, 780, 3855, 272, 4369, 4641, 5185, 6273, 288, 4626, 4386, 6210, 5250, 816, 13107, 15555, 320, 5140, 6180, 4420, 4740, 1360, 21845

Offset: 0

Tilman Piesk, Aug 08 2013

nonn

Each entry of this sequence represents the same small equivalence class (sec) of Boolean functions as the corresponding entry of A190939. While A190939 represents each sec by the unique odd number among the numeric values of its functions, this sequence represents each sec by the smallest among these numbers (as an entry of A227722).
All big equivalence classes (bec) of Boolean functions are also small equivalence classes. So all entries in the sequence of sona-becs (A227960) are also in this sequence of sona-secs.
This sequence takes its order from A190939, so it is not monotonic. Thus it is not a subsequence of A227722, and does not contain A227960 as a subsequence.
First entries: 1, 3, 5, 6, 15, 17, 18, 51, 20, 85, 105, 24, 102, 90, 60, 255.
First entries in numerical order: 1, 3, 5, 6, 15, 17, 18, 20, 24, 51, 60, 85, 90, 102, 105, 255.

			A190939(3) = 9. 9 belongs to the sec A227722(4) = 6. So a(3) = 6.
A190939(8) = 65. 65 belongs to the sec A227722(10) = 20. So a(8) = 20.

Tilman Piesk, Table of n, a(n) for n = 0..2824
Tilman Piesk, Table of n, A190939(n), a(n) for n = 0..2824 and the same with a graphical explanation for n = 0..66
Tilman Piesk, Subgroups of nimber addition (Wikiversity)
Tilman Piesk, Small equivalence classes of Boolean functions

A198260 Runs of 1s in binary strings corresponding to subgroups of nimber addition.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 2, 2, 4, 3, 2, 3, 4, 2, 1, 2, 2, 2, 2, 4, 3, 2, 4, 4, 2, 2, 2, 4, 4, 4, 3, 2, 4, 4, 4, 4, 2, 4, 3, 2, 4, 4, 4, 4, 4, 8, 7, 3, 5, 6, 2, 3, 4, 4, 4, 4, 6, 5, 4, 7, 8, 2, 3, 4, 2, 1, 2, 2, 2, 2, 4, 3, 2, 4, 4, 2, 2, 2, 4, 4, 4, 3, 2, 4, 4, 4, 4, 2, 4, 3, 2, 4, 4

Offset: 0

Tilman Piesk, Oct 22 2011

nonn

Counts the runs of consecutive ones in the binary representation of A190939. All positive integers appear in this sequence.

The first A006116(n) entries contain all integers from 1 to 2^(n-1). E.g. the first 67 entries contain all integers from 1 to 8. How often each integer appears is counted by the triangle A227961.

Tilman Piesk, Table of n, a(n) for n = 0..2824

Cf. A190939 (subgroups of nimber addition interpreted as binary numbers).
Cf. A069010 (runs of 1s).
Cf. A227961 (corresponding triangle).

a(n) = A069010(A190939(n)).

Changed offset to 0 as in A190939 by Tilman Piesk, Jan 25 2012

Significant edit by Tilman Piesk, Aug 01 2013

A227962 Triangle of permutations that assign sona-becs (A227960) to their complements.

Original entry on oeis.org

0, 1, 0, 3, 1, 2, 0, 7, 3, 5, 1, 6, 2, 4, 0, 15, 7, 12, 3, 13, 5, 9, 1, 14, 6, 10, 11, 2, 4, 8, 0, 31, 15, 26, 7, 28, 12, 20, 3, 29, 13, 22, 23, 5, 9, 17, 1, 30, 14, 24, 25, 6, 27, 10, 11, 18, 19, 2, 21, 4, 8, 16, 0

Offset: 0

Tilman Piesk, Aug 04 2013

Subgroups of nimber addition (sona, A190939) have complements (defined using their Walsh spectrum). All sona in the same sona-bec (A227960) have complements in a unique sona-bec, which thus can be called its complement.
The permutation in row n of this triangle assigns complementary sona-becs of size 2^n to each other. (It is thus self-inverse.)
Even rows contain fixed points, because some sona-becs with weight 2^(n/2) are their own complements. E.g., in row 4 the fixed points are 3, 5, 10 and 11.
Each row contains the row before as a subsequence.
0 is always complement with A076766(n)-1, so each row ends with 0, and the left column is A076766-1 (not A000225).

			T(4,1)=7 and T(4,7)=1, so 1 and 7 are complements for n=4.
T(4,3)=3, so 3 is its own complement for n=4.
Triangle begins:
    k = 0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
n
0       0
1       1   0
2       3   1   2   0
3       7   3   5   1   6   2   4   0
4      15   7  12   3  13   5   9   1  14   6  10  11   2   4   8   0

Tilman Piesk, Rows 0...7, flattened
Tilman Piesk, Rows 0...7 (the same with emphasis on subsequences)
Tilman Piesk, Complement pairs for n=0...7
Tilman Piesk, Graphic for n=4, complements are symmetric to each other
Tilman Piesk, Subgroups of nimber addition (Wikiversity)

A359527 Nonnegative numbers k such that if 2^i and 2^j appear in the binary expansion of k, then 2^(i OR j) also appears in the binary expansion of k (where OR denotes the bitwise OR operator).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 32, 33, 34, 35, 48, 49, 50, 51, 64, 65, 68, 69, 80, 81, 84, 85, 128, 129, 130, 131, 132, 133, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 152, 153, 160, 161, 162, 163, 164, 165, 168, 169, 170, 171

Offset: 1

Rémy Sigrist, Jan 04 2023

Equivalently, numbers whose binary expansions encode union-closed finite sets of finite sets of nonnegative integers:
- the encoding is based on a double application of A133457,
- for example: 11 -> {0, 1, 3} -> {{}, {0}, {0, 1}},
- a union-closed set f satisfies: for any i and j in f, the union of i and j belongs to f.
For any k >= 0, 2*k belongs to the sequence iff 2*k+1 belongs to the sequence.
This sequence has similarities with A190939; here we consider the bitwise OR operator, there the bitwise XOR operator.
This sequence is infinite as it contains the powers of 2.

			The first terms, alongside the corresponding union-closed sets, are:
  n     a(n)   Union-closed set
  ----  -----  ----------------------
     1      0  {}
     2      1  {{}}
     3      2  {{0}}
     4      3  {{}, {0}}
     5      4  {{1}}
     6      5  {{}, {1}}
     7      8  {{0, 1}}
     8      9  {{}, {0, 1}}
     9     10  {{0}, {0, 1}}
    10     11  {{}, {0}, {0, 1}}
    11     12  {{1}, {0, 1}}
    12     13  {{}, {1}, {0, 1}}
    13     14  {{0}, {1}, {0, 1}}
    14     15  {{}, {0}, {1}, {0, 1}}
    15     16  {{2}}
    16     17  {{}, {2}}
    17     32  {{0, 2}}

Cf. A133457, A190939 (XOR analog), A359528 (AND analog).

is(n) = { my (b=vector(hammingweight(n))); for (i=1, #b, n -= 2^b[i] = valuation(n,2)); setbinop(bitor, b)==b }

A359528 Nonnegative numbers k such that if 2^i and 2^j appear in the binary expansion of k, then 2^(i AND j) also appears in the binary expansion of k (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 34, 35, 37, 39, 42, 43, 47, 48, 49, 51, 53, 55, 59, 63, 64, 65, 67, 68, 69, 71, 76, 77, 79, 80, 81, 83, 85, 87, 93, 95, 112, 113, 115, 117, 119, 127, 128, 129, 130, 131

Offset: 1

Rémy Sigrist, Jan 04 2023

Equivalently, numbers whose binary expansions encode intersection-closed finite sets of finite sets of nonnegative integers:
- the encoding is based on a double application of A133457,
- for example: 11 -> {0, 1, 3} -> {{}, {0}, {0, 1}},
- an intersection-closed set f satisfies: for any i and j in f, the intersection of i and j belongs to f.
For any k >= 0, if 2*k belongs to the sequence then 2*k+1 belongs to the sequence.
This sequence has similarities with A190939; here we consider the bitwise AND operator, there the bitwise XOR operator.
This sequence is infinite as it contains the powers of 2.

			The first terms, alongside the corresponding intersection-closed sets, are:
  n     a(n)   Intersection-closed set
  ----  -----  -----------------------
     0      0  {}
     1      1  {{}}
     2      2  {{0}}
     3      3  {{}, {0}}
     4      4  {{1}}
     5      5  {{}, {1}}
     6      7  {{}, {0}, {1}}
     7      8  {{0, 1}}
     8      9  {{}, {0, 1}}
     9     10  {{0}, {0, 1}}
    10     11  {{}, {0}, {0, 1}}
    11     12  {{1}, {0, 1}}
    12     13  {{}, {1}, {0, 1}}
    13     15  {{}, {0}, {1}, {0, 1}}
    14     16  {{2}}
    15     17  {{}, {2}}
    16     19  {{}, {0}, {2}}
    17     21  {{}, {1}, {2}}

Index entries for sequences related to binary expansion of n

Cf. A133457, A190939 (XOR analog), A359527 (OR analog).

is(n) = { my (b=vector(hammingweight(n))); for (i=1, #b, n -= 2^b[i] = valuation(n,2)); setbinop(bitand, b)==b }

cp's OEIS Frontend

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

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A227963 Small equivalence classes (A227722) of subgroups of nimber addition (A190939).

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A198260 Runs of 1s in binary strings corresponding to subgroups of nimber addition.

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A227962 Triangle of permutations that assign sona-becs (A227960) to their complements.

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A359527 Nonnegative numbers k such that if 2^i and 2^j appear in the binary expansion of k, then 2^(i OR j) also appears in the binary expansion of k (where OR denotes the bitwise OR operator).

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A359528 Nonnegative numbers k such that if 2^i and 2^j appear in the binary expansion of k, then 2^(i AND j) also appears in the binary expansion of k (where AND denotes the bitwise AND operator).

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Comments

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PARI