proposed

approved

editing

proposed

aQ[n_]:=n==1||Length[Select[PrimePi/@First/@If[n==1, {}, FactorInteger[n]], , aQ]]==1;

Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331912, A331915, A331916.

allocated for Gus WisemanLexicographically earliest sequence containing 1 and all positive integers that have exactly one distinct prime index already in the sequence.

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, 65, 67, 74, 81, 82, 83, 86, 87, 91, 94, 97, 101, 103, 104, 111, 116, 117, 121, 122, 123, 125, 127, 128, 129, 131, 141, 142, 143, 145, 146, 148, 158, 164, 167, 172, 178

1,2

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.

The sequence of terms together with their prime indices begins:

1: {} 52: {1,1,6} 116: {1,1,10}

2: {1} 53: {16} 117: {2,2,6}

3: {2} 58: {1,10} 121: {5,5}

4: {1,1} 59: {17} 122: {1,18}

5: {3} 64: {1,1,1,1,1,1} 123: {2,13}

7: {4} 65: {3,6} 125: {3,3,3}

8: {1,1,1} 67: {19} 127: {31}

9: {2,2} 74: {1,12} 128: {1,1,1,1,1,1,1}

11: {5} 81: {2,2,2,2} 129: {2,14}

16: {1,1,1,1} 82: {1,13} 131: {32}

17: {7} 83: {23} 141: {2,15}

19: {8} 86: {1,14} 142: {1,20}

23: {9} 87: {2,10} 143: {5,6}

25: {3,3} 91: {4,6} 145: {3,10}

26: {1,6} 94: {1,15} 146: {1,21}

27: {2,2,2} 97: {25} 148: {1,1,12}

31: {11} 101: {26} 158: {1,22}

32: {1,1,1,1,1} 103: {27} 164: {1,1,13}

39: {2,6} 104: {1,1,1,6} 167: {39}

49: {4,4} 111: {2,12} 172: {1,1,14}

aQ[n_]:=n==1||Length[Select[PrimePi/@First/@If[n==1, {}, FactorInteger[n]], aQ]]==1;

Select[Range[200], aQ]

Contains all prime powers A000961.

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 exactly one prime index in S are A331785.

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

Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331912, A331915, A331916.

allocated

nonn

Gus Wiseman, Feb 01 2020

approved

editing

allocated for Gus Wiseman

allocated

approved