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%I #8 Feb 02 2020 09:03:59
%S 1,2,3,5,11,14,21,26,31,34,35,38,39,43,46,51,57,58,65,69,73,74,77,82,
%T 85,87,94,95,98,101,106,111,115,118,122,123,127,134,139,141,142,143,
%U 145,147,149,158,159,163,166,167,177,178,182,183,185,187,191,194,199
%N Lexicographically earliest sequence containing 1 and all positive integers with exactly one prime index already in the sequence, counting multiplicity.
%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: {} 73: {21} 142: {1,20} 205: {3,13}
%e 2: {1} 74: {1,12} 143: {5,6} 206: {1,27}
%e 3: {2} 77: {4,5} 145: {3,10} 209: {5,8}
%e 5: {3} 82: {1,13} 147: {2,4,4} 213: {2,20}
%e 11: {5} 85: {3,7} 149: {35} 214: {1,28}
%e 14: {1,4} 87: {2,10} 158: {1,22} 217: {4,11}
%e 21: {2,4} 94: {1,15} 159: {2,16} 218: {1,29}
%e 26: {1,6} 95: {3,8} 163: {38} 226: {1,30}
%e 31: {11} 98: {1,4,4} 166: {1,23} 233: {51}
%e 34: {1,7} 101: {26} 167: {39} 235: {3,15}
%e 35: {3,4} 106: {1,16} 177: {2,17} 237: {2,22}
%e 38: {1,8} 111: {2,12} 178: {1,24} 238: {1,4,7}
%e 39: {2,6} 115: {3,9} 182: {1,4,6} 245: {3,4,4}
%e 43: {14} 118: {1,17} 183: {2,18} 249: {2,23}
%e 46: {1,9} 122: {1,18} 185: {3,12} 253: {5,9}
%e 51: {2,7} 123: {2,13} 187: {5,7} 262: {1,32}
%e 57: {2,8} 127: {31} 191: {43} 265: {3,16}
%e 58: {1,10} 134: {1,19} 194: {1,25} 266: {1,4,8}
%e 65: {3,6} 139: {34} 199: {46} 267: {2,24}
%e 69: {2,9} 141: {2,15} 201: {2,19} 269: {57}
%e For example, the prime indices of 77 are {4,5}, of which only 5 is in the sequence, so 77 is in the sequence.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t aQ[n_]:=n==1||Length[Select[primeMS[n],aQ]]==1;
%t Select[Range[100],aQ]
%Y Closed under A000040.
%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 at most one distinct prime index in S are A331912.
%Y Numbers S with exactly one distinct prime index in S are A331913.
%Y Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331915, A331916.
%K nonn
%O 1,2
%A _Gus Wiseman_, Feb 01 2020