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A331785
Lexicographically earliest sequence containing 1 and all positive integers with exactly one prime index already in the sequence, counting multiplicity.
8
1, 2, 3, 5, 11, 14, 21, 26, 31, 34, 35, 38, 39, 43, 46, 51, 57, 58, 65, 69, 73, 74, 77, 82, 85, 87, 94, 95, 98, 101, 106, 111, 115, 118, 122, 123, 127, 134, 139, 141, 142, 143, 145, 147, 149, 158, 159, 163, 166, 167, 177, 178, 182, 183, 185, 187, 191, 194, 199
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.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {} 73: {21} 142: {1,20} 205: {3,13}
2: {1} 74: {1,12} 143: {5,6} 206: {1,27}
3: {2} 77: {4,5} 145: {3,10} 209: {5,8}
5: {3} 82: {1,13} 147: {2,4,4} 213: {2,20}
11: {5} 85: {3,7} 149: {35} 214: {1,28}
14: {1,4} 87: {2,10} 158: {1,22} 217: {4,11}
21: {2,4} 94: {1,15} 159: {2,16} 218: {1,29}
26: {1,6} 95: {3,8} 163: {38} 226: {1,30}
31: {11} 98: {1,4,4} 166: {1,23} 233: {51}
34: {1,7} 101: {26} 167: {39} 235: {3,15}
35: {3,4} 106: {1,16} 177: {2,17} 237: {2,22}
38: {1,8} 111: {2,12} 178: {1,24} 238: {1,4,7}
39: {2,6} 115: {3,9} 182: {1,4,6} 245: {3,4,4}
43: {14} 118: {1,17} 183: {2,18} 249: {2,23}
46: {1,9} 122: {1,18} 185: {3,12} 253: {5,9}
51: {2,7} 123: {2,13} 187: {5,7} 262: {1,32}
57: {2,8} 127: {31} 191: {43} 265: {3,16}
58: {1,10} 134: {1,19} 194: {1,25} 266: {1,4,8}
65: {3,6} 139: {34} 199: {46} 267: {2,24}
69: {2,9} 141: {2,15} 201: {2,19} 269: {57}
For example, the prime indices of 77 are {4,5}, of which only 5 is in the sequence, so 77 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
aQ[n_]:=n==1||Length[Select[primeMS[n], aQ]]==1;
Select[Range[100], aQ]
CROSSREFS
Closed under A000040.
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 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: A215354 A129201 A137692 * A249684 A275913 A004680
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 01 2020
STATUS
approved