A331912 - OEIS

%I #5 Feb 02 2020 09:04:09

%S 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,

%T 49,52,53,58,59,61,64,65,67,71,73,74,79,81,83,86,87,89,91,94,97,101,

%U 103,104,107,109,111,113,116,117,121,122,125,127,128,129,131,137

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

%C 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.

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

%H Gus Wiseman, <a href="/A331912/a331912.png">Plot of A331912(n)/A331784(n) for n = 1..3729.</a>

%e The sequence of terms together with their prime indices begins:

%e     1: {}              37: {12}              86: {1,14}

%e     2: {1}             39: {2,6}             87: {2,10}

%e     3: {2}             41: {13}              89: {24}

%e     4: {1,1}           43: {14}              91: {4,6}

%e     5: {3}             47: {15}              94: {1,15}

%e     7: {4}             49: {4,4}             97: {25}

%e     8: {1,1,1}         52: {1,1,6}          101: {26}

%e     9: {2,2}           53: {16}             103: {27}

%e    11: {5}             58: {1,10}           104: {1,1,1,6}

%e    13: {6}             59: {17}             107: {28}

%e    16: {1,1,1,1}       61: {18}             109: {29}

%e    17: {7}             64: {1,1,1,1,1,1}    111: {2,12}

%e    19: {8}             65: {3,6}            113: {30}

%e    23: {9}             67: {19}             116: {1,1,10}

%e    25: {3,3}           71: {20}             117: {2,2,6}

%e    26: {1,6}           73: {21}             121: {5,5}

%e    27: {2,2,2}         74: {1,12}           122: {1,18}

%e    29: {10}            79: {22}             125: {3,3,3}

%e    31: {11}            81: {2,2,2,2}        127: {31}

%e    32: {1,1,1,1,1}     83: {23}             128: {1,1,1,1,1,1,1}

%e 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.

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

%t Select[Range[100],aQ]

%Y Contains all prime powers A000961.

%Y Numbers S without all prime indices in S are A324694.

%Y Numbers S without any prime indices in S are A324695.

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

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

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 01 2020