A359528 - OEIS

%I #7 Jan 07 2023 13:05:26

%S 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 35,37,39,42,43,47,48,49,51,53,55,59,63,64,65,67,68,69,71,76,77,79,80,

%U 81,83,85,87,93,95,112,113,115,117,119,127,128,129,130,131

%N 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 Equivalently, numbers whose binary expansions encode intersection-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 - an intersection-closed set f satisfies: for any i and j in f, the intersection of i and j belongs to f.

%C For any k >= 0, if 2*k belongs to the sequence then 2*k+1 belongs to the sequence.

%C This sequence has similarities with A190939; here we consider the bitwise AND operator, there the bitwise XOR operator.

%C This sequence is infinite as it contains the powers of 2.

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%e The first terms, alongside the corresponding intersection-closed sets, are:

%e   n     a(n)   Intersection-closed set

%e   ----  -----  -----------------------

%e      0      0  {}

%e      1      1  {{}}

%e      2      2  {{0}}

%e      3      3  {{}, {0}}

%e      4      4  {{1}}

%e      5      5  {{}, {1}}

%e      6      7  {{}, {0}, {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     15  {{}, {0}, {1}, {0, 1}}

%e     14     16  {{2}}

%e     15     17  {{}, {2}}

%e     16     19  {{}, {0}, {2}}

%e     17     21  {{}, {1}, {2}}

%o (PARI) is(n) = { my (b=vector(hammingweight(n))); for (i=1, #b, n -= 2^b[i] = valuation(n,2)); setbinop(bitand, b)==b }

%Y Cf. A133457, A190939 (XOR analog), A359527 (OR analog).

%K nonn,base

%O 1,3

%A _Rémy Sigrist_, Jan 04 2023