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).

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, 35, 37, 39, 42, 43, 47, 48, 49, 51, 53, 55, 59, 63, 64, 65, 67, 68, 69, 71, 76, 77, 79, 80, 81, 83, 85, 87, 93, 95, 112, 113, 115, 117, 119, 127, 128, 129, 130, 131

Offset: 1

Author

Rémy Sigrist, Jan 04 2023

Keywords

• nonn
• base

Comments

Equivalently, numbers whose binary expansions encode intersection-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}},

- an intersection-closed set f satisfies: for any i and j in f, the intersection of i and j belongs to f.

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

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

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

Examples

			The first terms, alongside the corresponding intersection-closed sets, are:
  n     a(n)   Intersection-closed set
  ----  -----  -----------------------
     0      0  {}
     1      1  {{}}
     2      2  {{0}}
     3      3  {{}, {0}}
     4      4  {{1}}
     5      5  {{}, {1}}
     6      7  {{}, {0}, {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     15  {{}, {0}, {1}, {0, 1}}
    14     16  {{2}}
    15     17  {{}, {2}}
    16     19  {{}, {0}, {2}}
    17     21  {{}, {1}, {2}}
		

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

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

Programs

PARI

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