A331913 Lexicographically 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

Offset

1,2

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.

Links

Table of n, a(n) for n=1..61.

Example

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}

Mathematica

aQ[n_]:=n==1||Length[Select[PrimePi/@First/@FactorInteger[n], aQ]]==1;
Select[Range[200], aQ]

Crossrefs

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, A331915, A331916.

Sequence in context: A099627 A324841 A184155 * A318612 A318690 A302498

Adjacent sequences: A331910 A331911 A331912 * A331914 A331915 A331916

Keyword

nonn

Author

Gus Wiseman, Feb 01 2020

Status

approved