A331785 Lexicographically earliest sequence containing 1 and all positive integers with exactly one prime index already in the sequence, counting multiplicity.

1, 2, 3, 5, 11, 14, 21, 26, 31, 34, 35, 38, 39, 43, 46, 51, 57, 58, 65, 69, 73, 74, 77, 82, 85, 87, 94, 95, 98, 101, 106, 111, 115, 118, 122, 123, 127, 134, 139, 141, 142, 143, 145, 147, 149, 158, 159, 163, 166, 167, 177, 178, 182, 183, 185, 187, 191, 194, 199

Offset: 1

Author

Gus Wiseman, Feb 01 2020

Keywords

• nonn

Comments

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.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}         73: {21}       142: {1,20}     205: {3,13}
    2: {1}        74: {1,12}     143: {5,6}      206: {1,27}
    3: {2}        77: {4,5}      145: {3,10}     209: {5,8}
    5: {3}        82: {1,13}     147: {2,4,4}    213: {2,20}
   11: {5}        85: {3,7}      149: {35}       214: {1,28}
   14: {1,4}      87: {2,10}     158: {1,22}     217: {4,11}
   21: {2,4}      94: {1,15}     159: {2,16}     218: {1,29}
   26: {1,6}      95: {3,8}      163: {38}       226: {1,30}
   31: {11}       98: {1,4,4}    166: {1,23}     233: {51}
   34: {1,7}     101: {26}       167: {39}       235: {3,15}
   35: {3,4}     106: {1,16}     177: {2,17}     237: {2,22}
   38: {1,8}     111: {2,12}     178: {1,24}     238: {1,4,7}
   39: {2,6}     115: {3,9}      182: {1,4,6}    245: {3,4,4}
   43: {14}      118: {1,17}     183: {2,18}     249: {2,23}
   46: {1,9}     122: {1,18}     185: {3,12}     253: {5,9}
   51: {2,7}     123: {2,13}     187: {5,7}      262: {1,32}
   57: {2,8}     127: {31}       191: {43}       265: {3,16}
   58: {1,10}    134: {1,19}     194: {1,25}     266: {1,4,8}
   65: {3,6}     139: {34}       199: {46}       267: {2,24}
   69: {2,9}     141: {2,15}     201: {2,19}     269: {57}
For example, the prime indices of 77 are {4,5}, of which only 5 is in the sequence, so 77 is in the sequence.
		

Crossrefs

Closed under A000040.

Numbers S without all prime indices in S are A324694.

Numbers S without any prime indices in S are A324695.

Numbers S with at most one prime index in S are A331784.

Numbers S with at most one distinct prime index in S are A331912.

Numbers S with exactly one distinct prime index in S are A331913.

Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331915, A331916.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aQ[n_]:=n==1||Length[Select[primeMS[n],aQ]]==1;
Select[Range[100],aQ]