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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).
1
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
1,3
COMMENTS
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.
EXAMPLE
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}}
PROG
(PARI) is(n) = { my (b=vector(hammingweight(n))); for (i=1, #b, n -= 2^b[i] = valuation(n, 2)); setbinop(bitor, b)==b }
CROSSREFS
Cf. A133457, A190939 (XOR analog), A359528 (AND analog).
Sequence in context: A297163 A319024 A039260 * A039200 A039150 A008539
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Jan 04 2023
STATUS
approved