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Numbers whose binary indices are all squarefree.
9

%I #16 Oct 24 2024 16:17:34

%S 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,

%T 49,50,51,52,53,54,55,64,65,66,67,68,69,70,71,80,81,82,83,84,85,86,87,

%U 96,97,98,99,100,101,102,103,112,113,114,115,116,117,118,119,512

%N Numbers whose binary indices are all squarefree.

%C The complement first differs from A115419 in having 128.

%C 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.

%e The terms together with their binary expansions and binary indices begin:

%e 0: 0 ~ {}

%e 1: 1 ~ {1}

%e 2: 10 ~ {2}

%e 3: 11 ~ {1,2}

%e 4: 100 ~ {3}

%e 5: 101 ~ {1,3}

%e 6: 110 ~ {2,3}

%e 7: 111 ~ {1,2,3}

%e 16: 10000 ~ {5}

%e 17: 10001 ~ {1,5}

%e 18: 10010 ~ {2,5}

%e 19: 10011 ~ {1,2,5}

%e 20: 10100 ~ {3,5}

%e 21: 10101 ~ {1,3,5}

%e 22: 10110 ~ {2,3,5}

%e 23: 10111 ~ {1,2,3,5}

%e 32: 100000 ~ {6}

%e 33: 100001 ~ {1,6}

%e 34: 100010 ~ {2,6}

%e 35: 100011 ~ {1,2,6}

%e 36: 100100 ~ {3,6}

%e 37: 100101 ~ {1,3,6}

%e 38: 100110 ~ {2,3,6}

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t Select[Range[0,100],And@@SquareFreeQ/@bpe[#]&]

%o (Python)

%o from math import isqrt

%o from sympy import mobius

%o def A368533(n):

%o def f(x,n): return int(n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))

%o def A005117(n):

%o m, k = n, f(n,n)

%o while m != k: m, k = k, f(k,n)

%o return m

%o 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

%Y Set multipartitions: A049311, A050320, A089259, A116540.

%Y For prime indices instead of binary indices we have A302478.

%Y The case of prime binary indices is A326782.

%Y The case of squarefree product is A371289.

%Y For prime-power product we have A371290.

%Y For nonprime binary indices we have A371443, composite A371444.

%Y The semiprime case is A371453, squarefree case of A371454.

%Y A005117 lists squarefree numbers.

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

%Y A070939 gives length of binary expansion.

%Y A096111 gives product of binary indices.

%Y Cf. A000040, A001222, A087086, A296119, A326031, A367905, A368109, A371450.

%K nonn,base

%O 1,3

%A _Gus Wiseman_, Mar 23 2024