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A331912
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Lexicographically earliest sequence of positive integers that have at most one distinct prime index already in the sequence.
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13
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1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 26, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 52, 53, 58, 59, 61, 64, 65, 67, 71, 73, 74, 79, 81, 83, 86, 87, 89, 91, 94, 97, 101, 103, 104, 107, 109, 111, 113, 116, 117, 121, 122, 125, 127, 128, 129, 131, 137
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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.
Conjecture: a(n)/A331784(n) -> 1 as n -> infinity.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {} 37: {12} 86: {1,14}
2: {1} 39: {2,6} 87: {2,10}
3: {2} 41: {13} 89: {24}
4: {1,1} 43: {14} 91: {4,6}
5: {3} 47: {15} 94: {1,15}
7: {4} 49: {4,4} 97: {25}
8: {1,1,1} 52: {1,1,6} 101: {26}
9: {2,2} 53: {16} 103: {27}
11: {5} 58: {1,10} 104: {1,1,1,6}
13: {6} 59: {17} 107: {28}
16: {1,1,1,1} 61: {18} 109: {29}
17: {7} 64: {1,1,1,1,1,1} 111: {2,12}
19: {8} 65: {3,6} 113: {30}
23: {9} 67: {19} 116: {1,1,10}
25: {3,3} 71: {20} 117: {2,2,6}
26: {1,6} 73: {21} 121: {5,5}
27: {2,2,2} 74: {1,12} 122: {1,18}
29: {10} 79: {22} 125: {3,3,3}
31: {11} 81: {2,2,2,2} 127: {31}
32: {1,1,1,1,1} 83: {23} 128: {1,1,1,1,1,1,1}
For example, the prime indices of 117 are {2,2,6}, of which only 2 is already in the sequence, so 117 is in the sequence.
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MATHEMATICA
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aQ[n_]:=Length[Select[PrimePi/@First/@If[n==1, {}, FactorInteger[n]], aQ]]<=1;
Select[Range[100], aQ]
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CROSSREFS
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Numbers S without all prime indices in S are A324694.
Numbers S without any prime indices in S are A324695.
Numbers S with at most one prime index in S are A331784.
Numbers S with exactly one prime index in S are A331785.
Numbers S with exactly one distinct prime index in S are A331913.
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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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