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%I #6 Feb 02 2020 09:04:17
%S 1,2,3,4,5,7,8,9,11,16,17,19,23,25,26,27,31,32,39,49,52,53,58,59,64,
%T 65,67,74,81,82,83,86,87,91,94,97,101,103,104,111,116,117,121,122,123,
%U 125,127,128,129,131,141,142,143,145,146,148,158,164,167,172,178
%N Lexicographically earliest sequence containing 1 and all positive integers that have exactly one distinct prime index already in the sequence.
%C 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.
%e The sequence of terms together with their prime indices begins:
%e 1: {} 52: {1,1,6} 116: {1,1,10}
%e 2: {1} 53: {16} 117: {2,2,6}
%e 3: {2} 58: {1,10} 121: {5,5}
%e 4: {1,1} 59: {17} 122: {1,18}
%e 5: {3} 64: {1,1,1,1,1,1} 123: {2,13}
%e 7: {4} 65: {3,6} 125: {3,3,3}
%e 8: {1,1,1} 67: {19} 127: {31}
%e 9: {2,2} 74: {1,12} 128: {1,1,1,1,1,1,1}
%e 11: {5} 81: {2,2,2,2} 129: {2,14}
%e 16: {1,1,1,1} 82: {1,13} 131: {32}
%e 17: {7} 83: {23} 141: {2,15}
%e 19: {8} 86: {1,14} 142: {1,20}
%e 23: {9} 87: {2,10} 143: {5,6}
%e 25: {3,3} 91: {4,6} 145: {3,10}
%e 26: {1,6} 94: {1,15} 146: {1,21}
%e 27: {2,2,2} 97: {25} 148: {1,1,12}
%e 31: {11} 101: {26} 158: {1,22}
%e 32: {1,1,1,1,1} 103: {27} 164: {1,1,13}
%e 39: {2,6} 104: {1,1,1,6} 167: {39}
%e 49: {4,4} 111: {2,12} 172: {1,1,14}
%t aQ[n_]:=n==1||Length[Select[PrimePi/@First/@FactorInteger[n],aQ]]==1;
%t Select[Range[200],aQ]
%Y Contains all prime powers A000961.
%Y Numbers S without all prime indices in S are A324694.
%Y Numbers S without any prime indices in S are A324695.
%Y Numbers S with at most one prime index in S are A331784.
%Y Numbers S with exactly one prime index in S are A331785.
%Y Numbers S with at most one distinct prime index in S are A331912.
%Y Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331915, A331916.
%K nonn
%O 1,2
%A _Gus Wiseman_, Feb 01 2020