%I #5 Feb 02 2020 09:03:50
%S 1,2,3,5,7,11,13,14,17,19,21,23,26,29,31,35,37,38,39,41,43,46,47,49,
%T 53,57,58,59,61,65,67,69,71,73,74,77,79,83,87,89,91,94,95,97,98,101,
%U 103,106,107,109,111,113,115,119,122,127,131,133,137,139,141,142
%N Lexicographically earliest sequence of positive integers that have at most 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.
%C Conjecture: A331912(n)/a(n) -> 1 as n -> infinity.
%H Gus Wiseman, <a href="/A331784/a331784.png">Plot of A331912(n)/A331784(n) for n = 1..3729.</a>
%e The sequence of terms together with their prime indices begins:
%e 1: {} 43: {14} 91: {4,6} 141: {2,15}
%e 2: {1} 46: {1,9} 94: {1,15} 142: {1,20}
%e 3: {2} 47: {15} 95: {3,8} 143: {5,6}
%e 5: {3} 49: {4,4} 97: {25} 145: {3,10}
%e 7: {4} 53: {16} 98: {1,4,4} 147: {2,4,4}
%e 11: {5} 57: {2,8} 101: {26} 149: {35}
%e 13: {6} 58: {1,10} 103: {27} 151: {36}
%e 14: {1,4} 59: {17} 106: {1,16} 157: {37}
%e 17: {7} 61: {18} 107: {28} 158: {1,22}
%e 19: {8} 65: {3,6} 109: {29} 159: {2,16}
%e 21: {2,4} 67: {19} 111: {2,12} 161: {4,9}
%e 23: {9} 69: {2,9} 113: {30} 163: {38}
%e 26: {1,6} 71: {20} 115: {3,9} 167: {39}
%e 29: {10} 73: {21} 119: {4,7} 169: {6,6}
%e 31: {11} 74: {1,12} 122: {1,18} 173: {40}
%e 35: {3,4} 77: {4,5} 127: {31} 178: {1,24}
%e 37: {12} 79: {22} 131: {32} 179: {41}
%e 38: {1,8} 83: {23} 133: {4,8} 181: {42}
%e 39: {2,6} 87: {2,10} 137: {33} 182: {1,4,6}
%e 41: {13} 89: {24} 139: {34} 183: {2,18}
%e For example, the prime indices of 95 are {3,8}, of which only 3 is in the sequence, so 95 is in the sequence.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t aQ[n_]:=Length[Cases[primeMS[n],_?aQ]]<=1;
%t Select[Range[100],aQ]
%Y Contains all prime numbers 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 exactly one prime index in S are A331785.
%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, A331914.
%K nonn
%O 1,2
%A _Gus Wiseman_, Feb 01 2020