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A371291
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Numbers whose binary indices are connected, where two numbers are connected iff they have a common factor.
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9
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1, 2, 4, 8, 10, 16, 32, 34, 36, 38, 40, 42, 44, 46, 64, 128, 130, 136, 138, 160, 162, 164, 166, 168, 170, 172, 174, 256, 260, 288, 290, 292, 294, 296, 298, 300, 302, 416, 418, 420, 422, 424, 426, 428, 430, 512, 514, 520, 522, 528, 530, 536, 538, 544, 546, 548
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OFFSET
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1,2
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COMMENTS
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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.
The empty set is not considered connected.
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LINKS
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EXAMPLE
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The terms together with their binary expansions and binary indices begin:
1: 1 ~ {1}
2: 10 ~ {2}
4: 100 ~ {3}
8: 1000 ~ {4}
10: 1010 ~ {2,4}
16: 10000 ~ {5}
32: 100000 ~ {6}
34: 100010 ~ {2,6}
36: 100100 ~ {3,6}
38: 100110 ~ {2,3,6}
40: 101000 ~ {4,6}
42: 101010 ~ {2,4,6}
44: 101100 ~ {3,4,6}
46: 101110 ~ {2,3,4,6}
64: 1000000 ~ {7}
128: 10000000 ~ {8}
130: 10000010 ~ {2,8}
136: 10001000 ~ {4,8}
138: 10001010 ~ {2,4,8}
160: 10100000 ~ {6,8}
162: 10100010 ~ {2,6,8}
164: 10100100 ~ {3,6,8}
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MATHEMATICA
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csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 1000], Length[csm[prix/@bpe[#]]]==1&]
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CROSSREFS
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For prime indices of each prime index we have A305078.
For binary indices of each binary index we have A326749.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A070939 gives length of binary expansion.
A087086 lists numbers whose binary indices are pairwise indivisible.
A096111 gives product of binary indices.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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