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A359527
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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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1
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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}}
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PROG
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(PARI) is(n) = { my (b=vector(hammingweight(n))); for (i=1, #b, n -= 2^b[i] = valuation(n, 2)); setbinop(bitor, b)==b }
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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