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A331913 Lexicographically earliest sequence containing 1 and all positive integers that have exactly one distinct prime index already in the sequence.

%I A331913 #6 Feb 02 2020 09:04:17
%S A331913 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 A331913 65,67,74,81,82,83,86,87,91,94,97,101,103,104,111,116,117,121,122,123,
%U A331913 125,127,128,129,131,141,142,143,145,146,148,158,164,167,172,178
%N A331913 Lexicographically earliest sequence containing 1 and all positive integers that have exactly one distinct prime index already in the sequence.
%C A331913 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 A331913 The sequence of terms together with their prime indices begins:
%e A331913     1: {}              52: {1,1,6}          116: {1,1,10}
%e A331913     2: {1}             53: {16}             117: {2,2,6}
%e A331913     3: {2}             58: {1,10}           121: {5,5}
%e A331913     4: {1,1}           59: {17}             122: {1,18}
%e A331913     5: {3}             64: {1,1,1,1,1,1}    123: {2,13}
%e A331913     7: {4}             65: {3,6}            125: {3,3,3}
%e A331913     8: {1,1,1}         67: {19}             127: {31}
%e A331913     9: {2,2}           74: {1,12}           128: {1,1,1,1,1,1,1}
%e A331913    11: {5}             81: {2,2,2,2}        129: {2,14}
%e A331913    16: {1,1,1,1}       82: {1,13}           131: {32}
%e A331913    17: {7}             83: {23}             141: {2,15}
%e A331913    19: {8}             86: {1,14}           142: {1,20}
%e A331913    23: {9}             87: {2,10}           143: {5,6}
%e A331913    25: {3,3}           91: {4,6}            145: {3,10}
%e A331913    26: {1,6}           94: {1,15}           146: {1,21}
%e A331913    27: {2,2,2}         97: {25}             148: {1,1,12}
%e A331913    31: {11}           101: {26}             158: {1,22}
%e A331913    32: {1,1,1,1,1}    103: {27}             164: {1,1,13}
%e A331913    39: {2,6}          104: {1,1,1,6}        167: {39}
%e A331913    49: {4,4}          111: {2,12}           172: {1,1,14}
%t A331913 aQ[n_]:=n==1||Length[Select[PrimePi/@First/@FactorInteger[n],aQ]]==1;
%t A331913 Select[Range[200],aQ]
%Y A331913 Contains all prime powers A000961.
%Y A331913 Numbers S without all prime indices in S are A324694.
%Y A331913 Numbers S without any prime indices in S are A324695.
%Y A331913 Numbers S with at most one prime index in S are A331784.
%Y A331913 Numbers S with exactly one prime index in S are A331785.
%Y A331913 Numbers S with at most one distinct prime index in S are A331912.
%Y A331913 Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331915, A331916.
%K A331913 nonn
%O A331913 1,2
%A A331913 _Gus Wiseman_, Feb 01 2020