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A371293
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Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.
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4
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1, 2, 3, 6, 7, 22, 23, 32, 33, 48, 49, 86, 87, 112, 113, 516, 517, 580, 581, 1110, 1111, 1136, 1137, 1604, 1605, 5206, 5207, 5232, 5233, 5700, 5701, 8212, 8213, 9236, 9237, 13332, 13333, 16386, 16387, 16450, 16451, 17474, 17475, 21570, 21571, 24576, 24577
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OFFSET
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1,2
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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.
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LINKS
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FORMULA
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EXAMPLE
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The terms together with their prime indices of binary indices begin:
1: {{}}
2: {{1}}
3: {{},{1}}
6: {{1},{2}}
7: {{},{1},{2}}
22: {{1},{2},{3}}
23: {{},{1},{2},{3}}
32: {{1,2}}
33: {{},{1,2}}
48: {{3},{1,2}}
49: {{},{3},{1,2}}
86: {{1},{2},{3},{4}}
87: {{},{1},{2},{3},{4}}
112: {{3},{1,2},{4}}
113: {{},{3},{1,2},{4}}
516: {{2},{1,3}}
517: {{},{2},{1,3}}
580: {{2},{4},{1,3}}
581: {{},{2},{4},{1,3}}
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MATHEMATICA
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normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
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[1000], SquareFreeQ[Times @@ bpe[#]]&&normQ[Join@@prix/@bpe[#]]&]
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CROSSREFS
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Without the covering condition we have A371289.
Without squarefree product we have A371292.
Interchanging binary and prime indices gives A371448.
A000009 counts partitions covering initial interval, compositions A107429.
A011782 counts multisets covering an initial interval.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A131689 counts patterns by number of distinct parts.
A326701 lists BII-numbers of set partitions.
A368533 lists numbers with squarefree binary indices, prime indices A302478.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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