cp's OEIS Frontend

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, and {{2},{1,3}} is a set partition, it follows that 18 belongs to the sequence.

Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in exactly one way.

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary digits (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

a(n) is even if and only if n is in A062880. - Robert Israel, Oct 13 2020

A composition of n is a finite sequence of positive integers summing to n.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

From Gus Wiseman, Dec 29 2023: (Start)

These are numbers whose binary indices are all powers of 2, where a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, the terms together with their binary expansions and binary indices begin:

0: 0 ~ {}

1: 1 ~ {1}

2: 10 ~ {2}

3: 11 ~ {1,2}

8: 1000 ~ {4}

9: 1001 ~ {1,4}

10: 1010 ~ {2,4}

11: 1011 ~ {1,2,4}

128: 10000000 ~ {8}

129: 10000001 ~ {1,8}

130: 10000010 ~ {2,8}

131: 10000011 ~ {1,2,8}

136: 10001000 ~ {4,8}

137: 10001001 ~ {1,4,8}

138: 10001010 ~ {2,4,8}

139: 10001011 ~ {1,2,4,8}

For powers of 3 we have A368531.

(End)

Also numbers k of the form Sum_{b=0..h-1} 2^(i+j*b) for some h >= 0, i >= 0, j > 0 (in fact, h = A000120(k), and if k > 0, i = A007814(k)).

There is a simple bijection between the finite sets of nonnegative integers in arithmetic progression and the terms of this sequence: s -> Sum_{i in s} 2^i; the term 0 corresponds to the empty set.

For any n > 0, A054519(n) gives the numbers of terms with n+1 digits in binary representation.

For any n >= 0, n is in the sequence iff 2*n is in the sequence.

For any n > 0, A000695(a(n)) is in the sequence.

The first prime numbers in the sequence are: 2, 3, 5, 7, 17, 31, 73, 127, 257, 8191, 65537, 131071, 262657, 524287, ...

This sequence contains the following sequences: A000051, A000079, A000225, A000668, A002450, A019434, A023001, A048645.

For any k > 0, 2^k - 2, 2^k - 1, 2^k, 2^k + 1 and 2^k + 2 are in the sequence (e.g., 14, 15, 16, 17, and 18).

Every odd term is a binary palindrome (and thus belongs to A006995).

Odd terms are A064896. - Robert Israel, Nov 20 2017