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%I #9 Jan 07 2023 13:05:37
%S 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,
%T 65,68,69,80,81,84,85,128,129,130,131,132,133,136,137,138,139,140,141,
%U 142,143,144,145,152,153,160,161,162,163,164,165,168,169,170,171
%N 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).
%C Equivalently, numbers whose binary expansions encode union-closed finite sets of finite sets of nonnegative integers:
%C - the encoding is based on a double application of A133457,
%C - for example: 11 -> {0, 1, 3} -> {{}, {0}, {0, 1}},
%C - a union-closed set f satisfies: for any i and j in f, the union of i and j belongs to f.
%C For any k >= 0, 2*k belongs to the sequence iff 2*k+1 belongs to the sequence.
%C This sequence has similarities with A190939; here we consider the bitwise OR operator, there the bitwise XOR operator.
%C This sequence is infinite as it contains the powers of 2.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Union-closed_sets_conjecture">Union-closed sets conjecture</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e The first terms, alongside the corresponding union-closed sets, are:
%e n a(n) Union-closed set
%e ---- ----- ----------------------
%e 1 0 {}
%e 2 1 {{}}
%e 3 2 {{0}}
%e 4 3 {{}, {0}}
%e 5 4 {{1}}
%e 6 5 {{}, {1}}
%e 7 8 {{0, 1}}
%e 8 9 {{}, {0, 1}}
%e 9 10 {{0}, {0, 1}}
%e 10 11 {{}, {0}, {0, 1}}
%e 11 12 {{1}, {0, 1}}
%e 12 13 {{}, {1}, {0, 1}}
%e 13 14 {{0}, {1}, {0, 1}}
%e 14 15 {{}, {0}, {1}, {0, 1}}
%e 15 16 {{2}}
%e 16 17 {{}, {2}}
%e 17 32 {{0, 2}}
%o (PARI) is(n) = { my (b=vector(hammingweight(n))); for (i=1, #b, n -= 2^b[i] = valuation(n,2)); setbinop(bitor, b)==b }
%Y Cf. A133457, A190939 (XOR analog), A359528 (AND analog).
%K nonn,base
%O 1,3
%A _Rémy Sigrist_, Jan 04 2023
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