cp's OEIS Frontend

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).

%I A359527 #9 Jan 07 2023 13:05:37
%S A359527 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 A359527 65,68,69,80,81,84,85,128,129,130,131,132,133,136,137,138,139,140,141,
%U A359527 142,143,144,145,152,153,160,161,162,163,164,165,168,169,170,171
%N 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).
%C A359527 Equivalently, numbers whose binary expansions encode union-closed finite sets of finite sets of nonnegative integers:
%C A359527 - the encoding is based on a double application of A133457,
%C A359527 - for example: 11 -> {0, 1, 3} -> {{}, {0}, {0, 1}},
%C A359527 - a union-closed set f satisfies: for any i and j in f, the union of i and j belongs to f.
%C A359527 For any k >= 0, 2*k belongs to the sequence iff 2*k+1 belongs to the sequence.
%C A359527 This sequence has similarities with A190939; here we consider the bitwise OR operator, there the bitwise XOR operator.
%C A359527 This sequence is infinite as it contains the powers of 2.
%H A359527 Wikipedia, <a href="https://en.wikipedia.org/wiki/Union-closed_sets_conjecture">Union-closed sets conjecture</a>
%H A359527 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e A359527 The first terms, alongside the corresponding union-closed sets, are:
%e A359527   n     a(n)   Union-closed set
%e A359527   ----  -----  ----------------------
%e A359527      1      0  {}
%e A359527      2      1  {{}}
%e A359527      3      2  {{0}}
%e A359527      4      3  {{}, {0}}
%e A359527      5      4  {{1}}
%e A359527      6      5  {{}, {1}}
%e A359527      7      8  {{0, 1}}
%e A359527      8      9  {{}, {0, 1}}
%e A359527      9     10  {{0}, {0, 1}}
%e A359527     10     11  {{}, {0}, {0, 1}}
%e A359527     11     12  {{1}, {0, 1}}
%e A359527     12     13  {{}, {1}, {0, 1}}
%e A359527     13     14  {{0}, {1}, {0, 1}}
%e A359527     14     15  {{}, {0}, {1}, {0, 1}}
%e A359527     15     16  {{2}}
%e A359527     16     17  {{}, {2}}
%e A359527     17     32  {{0, 2}}
%o A359527 (PARI) is(n) = { my (b=vector(hammingweight(n))); for (i=1, #b, n -= 2^b[i] = valuation(n,2)); setbinop(bitor, b)==b }
%Y A359527 Cf. A133457, A190939 (XOR analog), A359528 (AND analog).
%K A359527 nonn,base
%O A359527 1,3
%A A359527 _Rémy Sigrist_, Jan 04 2023