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

1,3

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.

Links

Table of n, a(n) for n=1..65.

Index entries for sequences related to binary expansion of n

Example

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}}

Prog

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

Crossrefs

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

Sequence in context: A079298 A263715 A023055 * A365819 A359908 A230872

Adjacent sequences: A359525 A359526 A359527 * A359529 A359530 A359531

Keyword

nonn,base

Author

Rémy Sigrist, Jan 04 2023

Status

approved