A331784 Lexicographically earliest sequence of positive integers that have at most one prime index already in the sequence, counting multiplicity.

1, 2, 3, 5, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 35, 37, 38, 39, 41, 43, 46, 47, 49, 53, 57, 58, 59, 61, 65, 67, 69, 71, 73, 74, 77, 79, 83, 87, 89, 91, 94, 95, 97, 98, 101, 103, 106, 107, 109, 111, 113, 115, 119, 122, 127, 131, 133, 137, 139, 141, 142

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: A331912(n)/a(n) -> 1 as n -> infinity.

Links

Table of n, a(n) for n=1..62.

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

Example

The sequence of terms together with their prime indices begins:
    1: {}        43: {14}       91: {4,6}      141: {2,15}
    2: {1}       46: {1,9}      94: {1,15}     142: {1,20}
    3: {2}       47: {15}       95: {3,8}      143: {5,6}
    5: {3}       49: {4,4}      97: {25}       145: {3,10}
    7: {4}       53: {16}       98: {1,4,4}    147: {2,4,4}
   11: {5}       57: {2,8}     101: {26}       149: {35}
   13: {6}       58: {1,10}    103: {27}       151: {36}
   14: {1,4}     59: {17}      106: {1,16}     157: {37}
   17: {7}       61: {18}      107: {28}       158: {1,22}
   19: {8}       65: {3,6}     109: {29}       159: {2,16}
   21: {2,4}     67: {19}      111: {2,12}     161: {4,9}
   23: {9}       69: {2,9}     113: {30}       163: {38}
   26: {1,6}     71: {20}      115: {3,9}      167: {39}
   29: {10}      73: {21}      119: {4,7}      169: {6,6}
   31: {11}      74: {1,12}    122: {1,18}     173: {40}
   35: {3,4}     77: {4,5}     127: {31}       178: {1,24}
   37: {12}      79: {22}      131: {32}       179: {41}
   38: {1,8}     83: {23}      133: {4,8}      181: {42}
   39: {2,6}     87: {2,10}    137: {33}       182: {1,4,6}
   41: {13}      89: {24}      139: {34}       183: {2,18}
For example, the prime indices of 95 are {3,8}, of which only 3 is in the sequence, so 95 is in the sequence.

Mathematica

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

Crossrefs

Contains all prime numbers A000040.

Numbers S without all prime indices in S are A324694.

Numbers S without any prime indices in S are A324695.

Numbers S with exactly one prime index in S are A331785.

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, A331914.

Sequence in context: A074895 A371653 A298536 * A325162 A331635 A320055

Adjacent sequences: A331781 A331782 A331783 * A331785 A331786 A331787

Keyword

nonn

Author

Gus Wiseman, Feb 01 2020

Status

approved