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Numbers whose binary indices have squarefree product.
5

%I #11 Mar 28 2024 11:58:01

%S 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,

%T 69,70,71,80,81,82,83,84,85,86,87,96,97,112,113,512,513,516,517,576,

%U 577,580,581,1024,1025,1026,1027,1028,1029,1030,1031,1040,1041,1042

%N Numbers whose binary indices have squarefree product.

%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 48: 110000 ~ {5,6}

%e 49: 110001 ~ {1,5,6}

%e 64: 1000000 ~ {7}

%e 65: 1000001 ~ {1,7}

%e 66: 1000010 ~ {2,7}

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

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

%Y For prime instead of binary indices we have A302505.

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

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

%K nonn,base

%O 1,3

%A _Gus Wiseman_, Mar 25 2024