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A331784
Lexicographically earliest sequence of positive integers that have at most one prime index already in the sequence, counting multiplicity.
11
1, 2, 3, 5, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 35, 37, 38, 39, 41, 43, 46, 47, 49, 53, 57, 58, 59, 61, 65, 67, 69, 71, 73, 74, 77, 79, 83, 87, 89, 91, 94, 95, 97, 98, 101, 103, 106, 107, 109, 111, 113, 115, 119, 122, 127, 131, 133, 137, 139, 141, 142
OFFSET
1,2
COMMENTS
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: A331912(n)/a(n) -> 1 as n -> infinity.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {} 43: {14} 91: {4,6} 141: {2,15}
2: {1} 46: {1,9} 94: {1,15} 142: {1,20}
3: {2} 47: {15} 95: {3,8} 143: {5,6}
5: {3} 49: {4,4} 97: {25} 145: {3,10}
7: {4} 53: {16} 98: {1,4,4} 147: {2,4,4}
11: {5} 57: {2,8} 101: {26} 149: {35}
13: {6} 58: {1,10} 103: {27} 151: {36}
14: {1,4} 59: {17} 106: {1,16} 157: {37}
17: {7} 61: {18} 107: {28} 158: {1,22}
19: {8} 65: {3,6} 109: {29} 159: {2,16}
21: {2,4} 67: {19} 111: {2,12} 161: {4,9}
23: {9} 69: {2,9} 113: {30} 163: {38}
26: {1,6} 71: {20} 115: {3,9} 167: {39}
29: {10} 73: {21} 119: {4,7} 169: {6,6}
31: {11} 74: {1,12} 122: {1,18} 173: {40}
35: {3,4} 77: {4,5} 127: {31} 178: {1,24}
37: {12} 79: {22} 131: {32} 179: {41}
38: {1,8} 83: {23} 133: {4,8} 181: {42}
39: {2,6} 87: {2,10} 137: {33} 182: {1,4,6}
41: {13} 89: {24} 139: {34} 183: {2,18}
For example, the prime indices of 95 are {3,8}, of which only 3 is in the sequence, so 95 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
aQ[n_]:=Length[Cases[primeMS[n], _?aQ]]<=1;
Select[Range[100], aQ]
CROSSREFS
Contains all prime numbers A000040.
Numbers S without all prime indices in S are A324694.
Numbers S without any prime indices in S are A324695.
Numbers S with exactly one prime index in S are A331785.
Numbers S with at most one distinct prime index in S are A331912.
Numbers S with exactly one distinct prime index in S are A331913.
Sequence in context: A074895 A371653 A298536 * A325162 A331635 A320055
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 01 2020
STATUS
approved