cp's OEIS Frontend

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).

%I A359528 #7 Jan 07 2023 13:05:26
%S A359528 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,
%T A359528 35,37,39,42,43,47,48,49,51,53,55,59,63,64,65,67,68,69,71,76,77,79,80,
%U A359528 81,83,85,87,93,95,112,113,115,117,119,127,128,129,130,131
%N 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).
%C A359528 Equivalently, numbers whose binary expansions encode intersection-closed finite sets of finite sets of nonnegative integers:
%C A359528 - the encoding is based on a double application of A133457,
%C A359528 - for example: 11 -> {0, 1, 3} -> {{}, {0}, {0, 1}},
%C A359528 - an intersection-closed set f satisfies: for any i and j in f, the intersection of i and j belongs to f.
%C A359528 For any k >= 0, if 2*k belongs to the sequence then 2*k+1 belongs to the sequence.
%C A359528 This sequence has similarities with A190939; here we consider the bitwise AND operator, there the bitwise XOR operator.
%C A359528 This sequence is infinite as it contains the powers of 2.
%H A359528 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e A359528 The first terms, alongside the corresponding intersection-closed sets, are:
%e A359528   n     a(n)   Intersection-closed set
%e A359528   ----  -----  -----------------------
%e A359528      0      0  {}
%e A359528      1      1  {{}}
%e A359528      2      2  {{0}}
%e A359528      3      3  {{}, {0}}
%e A359528      4      4  {{1}}
%e A359528      5      5  {{}, {1}}
%e A359528      6      7  {{}, {0}, {1}}
%e A359528      7      8  {{0, 1}}
%e A359528      8      9  {{}, {0, 1}}
%e A359528      9     10  {{0}, {0, 1}}
%e A359528     10     11  {{}, {0}, {0, 1}}
%e A359528     11     12  {{1}, {0, 1}}
%e A359528     12     13  {{}, {1}, {0, 1}}
%e A359528     13     15  {{}, {0}, {1}, {0, 1}}
%e A359528     14     16  {{2}}
%e A359528     15     17  {{}, {2}}
%e A359528     16     19  {{}, {0}, {2}}
%e A359528     17     21  {{}, {1}, {2}}
%o A359528 (PARI) is(n) = { my (b=vector(hammingweight(n))); for (i=1, #b, n -= 2^b[i] = valuation(n,2)); setbinop(bitand, b)==b }
%Y A359528 Cf. A133457, A190939 (XOR analog), A359527 (OR analog).
%K A359528 nonn,base
%O A359528 1,3
%A A359528 _Rémy Sigrist_, Jan 04 2023