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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.

%I A331785 #8 Feb 02 2020 09:03:59
%S A331785 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 A331785 85,87,94,95,98,101,106,111,115,118,122,123,127,134,139,141,142,143,
%U A331785 145,147,149,158,159,163,166,167,177,178,182,183,185,187,191,194,199
%N A331785 Lexicographically earliest sequence containing 1 and all positive integers with exactly one prime index already in the sequence, counting multiplicity.
%C A331785 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 A331785 The sequence of terms together with their prime indices begins:
%e A331785     1: {}         73: {21}       142: {1,20}     205: {3,13}
%e A331785     2: {1}        74: {1,12}     143: {5,6}      206: {1,27}
%e A331785     3: {2}        77: {4,5}      145: {3,10}     209: {5,8}
%e A331785     5: {3}        82: {1,13}     147: {2,4,4}    213: {2,20}
%e A331785    11: {5}        85: {3,7}      149: {35}       214: {1,28}
%e A331785    14: {1,4}      87: {2,10}     158: {1,22}     217: {4,11}
%e A331785    21: {2,4}      94: {1,15}     159: {2,16}     218: {1,29}
%e A331785    26: {1,6}      95: {3,8}      163: {38}       226: {1,30}
%e A331785    31: {11}       98: {1,4,4}    166: {1,23}     233: {51}
%e A331785    34: {1,7}     101: {26}       167: {39}       235: {3,15}
%e A331785    35: {3,4}     106: {1,16}     177: {2,17}     237: {2,22}
%e A331785    38: {1,8}     111: {2,12}     178: {1,24}     238: {1,4,7}
%e A331785    39: {2,6}     115: {3,9}      182: {1,4,6}    245: {3,4,4}
%e A331785    43: {14}      118: {1,17}     183: {2,18}     249: {2,23}
%e A331785    46: {1,9}     122: {1,18}     185: {3,12}     253: {5,9}
%e A331785    51: {2,7}     123: {2,13}     187: {5,7}      262: {1,32}
%e A331785    57: {2,8}     127: {31}       191: {43}       265: {3,16}
%e A331785    58: {1,10}    134: {1,19}     194: {1,25}     266: {1,4,8}
%e A331785    65: {3,6}     139: {34}       199: {46}       267: {2,24}
%e A331785    69: {2,9}     141: {2,15}     201: {2,19}     269: {57}
%e A331785 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 A331785 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A331785 aQ[n_]:=n==1||Length[Select[primeMS[n],aQ]]==1;
%t A331785 Select[Range[100],aQ]
%Y A331785 Closed under A000040.
%Y A331785 Numbers S without all prime indices in S are A324694.
%Y A331785 Numbers S without any prime indices in S are A324695.
%Y A331785 Numbers S with at most one prime index in S are A331784.
%Y A331785 Numbers S with at most one distinct prime index in S are A331912.
%Y A331785 Numbers S with exactly one distinct prime index in S are A331913.
%Y A331785 Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331915, A331916.
%K A331785 nonn
%O A331785 1,2
%A A331785 _Gus Wiseman_, Feb 01 2020