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).

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

Offset: 1

Author

Rémy Sigrist, Jan 04 2023

Keywords

• nonn
• base

Comments

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.

Examples

			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}}
		

Links

• Wikipedia, Union-closed sets conjecture
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A133457, A190939 (XOR analog), A359528 (AND analog).

Programs

PARI

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