A371289 Numbers whose binary indices have squarefree product.

0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 48, 49, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87, 96, 97, 112, 113, 512, 513, 516, 517, 576, 577, 580, 581, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1040, 1041, 1042

Offset

1,3

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Links

Table of n, a(n) for n=1..59.

Example

The terms together with their binary expansions and binary indices begin:
     0:              0 ~ {}
     1:              1 ~ {1}
     2:             10 ~ {2}
     3:             11 ~ {1,2}
     4:            100 ~ {3}
     5:            101 ~ {1,3}
     6:            110 ~ {2,3}
     7:            111 ~ {1,2,3}
    16:          10000 ~ {5}
    17:          10001 ~ {1,5}
    18:          10010 ~ {2,5}
    19:          10011 ~ {1,2,5}
    20:          10100 ~ {3,5}
    21:          10101 ~ {1,3,5}
    22:          10110 ~ {2,3,5}
    23:          10111 ~ {1,2,3,5}
    32:         100000 ~ {6}
    33:         100001 ~ {1,6}
    48:         110000 ~ {5,6}
    49:         110001 ~ {1,5,6}
    64:        1000000 ~ {7}
    65:        1000001 ~ {1,7}
    66:        1000010 ~ {2,7}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 100], SquareFreeQ[Times@@bpe[#]]&]

Crossrefs

For prime instead of binary indices we have A302505.

For squarefree parts we have A368533, for prime indices A302478.

A005117 lists squarefree numbers.

A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A325118, A326782, A371290, A371291, A371292, A371293, A371443, A371446, A371448, A371449, A371452, A371453.

Sequence in context: A161673 A368533 A309126 * A004836 A039030 A162231

Adjacent sequences: A371286 A371287 A371288 * A371290 A371291 A371292

Keyword

nonn,base

Author

Gus Wiseman, Mar 25 2024

Status

approved