A368533 Numbers whose binary indices are all squarefree.

0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 48, 49, 50, 51, 52, 53, 54, 55, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87, 96, 97, 98, 99, 100, 101, 102, 103, 112, 113, 114, 115, 116, 117, 118, 119, 512

Offset: 1

Author

Gus Wiseman, Mar 23 2024

Keywords

• nonn
• base

Comments

The complement first differs from A115419 in having 128.

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.

Examples

			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}
   34:  100010 ~ {2,6}
   35:  100011 ~ {1,2,6}
   36:  100100 ~ {3,6}
   37:  100101 ~ {1,3,6}
   38:  100110 ~ {2,3,6}
		

Crossrefs

Set multipartitions: A049311, A050320, A089259, A116540.

For prime indices instead of binary indices we have A302478.

The case of prime binary indices is A326782.

The case of squarefree product is A371289.

For prime-power product we have A371290.

For nonprime binary indices we have A371443, composite A371444.

The semiprime case is A371453, squarefree case of A371454.

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. A000040, A001222, A087086, A296119, A326031, A367905, A368109, A371450.

Programs

Mathematica

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

Python

from math import isqrt
from sympy import mobius
def A368533(n):
    def f(x,n): return int(n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    def A005117(n):
        m, k = n, f(n,n)
        while m != k: m, k = k, f(k,n)
        return m
    return sum(1<<A005117(i)-1 for i, j in enumerate(bin(n-1)[:1:-1],1) if j=='1') # Chai Wah Wu, Oct 24 2024