A331912 Lexicographically earliest sequence of positive integers that have at most one distinct prime index already in the sequence.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 26, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 52, 53, 58, 59, 61, 64, 65, 67, 71, 73, 74, 79, 81, 83, 86, 87, 89, 91, 94, 97, 101, 103, 104, 107, 109, 111, 113, 116, 117, 121, 122, 125, 127, 128, 129, 131, 137

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.

Conjecture: a(n)/A331784(n) -> 1 as n -> infinity.

Links

Table of n, a(n) for n=1..63.

Gus Wiseman, Plot of A331912(n)/A331784(n) for n = 1..3729.

Example

The sequence of terms together with their prime indices begins:
    1: {}              37: {12}              86: {1,14}
    2: {1}             39: {2,6}             87: {2,10}
    3: {2}             41: {13}              89: {24}
    4: {1,1}           43: {14}              91: {4,6}
    5: {3}             47: {15}              94: {1,15}
    7: {4}             49: {4,4}             97: {25}
    8: {1,1,1}         52: {1,1,6}          101: {26}
    9: {2,2}           53: {16}             103: {27}
   11: {5}             58: {1,10}           104: {1,1,1,6}
   13: {6}             59: {17}             107: {28}
   16: {1,1,1,1}       61: {18}             109: {29}
   17: {7}             64: {1,1,1,1,1,1}    111: {2,12}
   19: {8}             65: {3,6}            113: {30}
   23: {9}             67: {19}             116: {1,1,10}
   25: {3,3}           71: {20}             117: {2,2,6}
   26: {1,6}           73: {21}             121: {5,5}
   27: {2,2,2}         74: {1,12}           122: {1,18}
   29: {10}            79: {22}             125: {3,3,3}
   31: {11}            81: {2,2,2,2}        127: {31}
   32: {1,1,1,1,1}     83: {23}             128: {1,1,1,1,1,1,1}
For example, the prime indices of 117 are {2,2,6}, of which only 2 is already in the sequence, so 117 is in the sequence.

Mathematica

aQ[n_]:=Length[Select[PrimePi/@First/@If[n==1, {}, FactorInteger[n]], aQ]]<=1;
Select[Range[100], 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 exactly one distinct prime index in S are A331913.

Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331914.

Sequence in context: A302040 A302036 A357864 * A326848 A328957 A030230

Adjacent sequences: A331909 A331910 A331911 * A331913 A331914 A331915

Keyword

nonn

Author

Gus Wiseman, Feb 01 2020

Status

approved