cp's OEIS Frontend

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

%I A331784 #5 Feb 02 2020 09:03:50
%S A331784 1,2,3,5,7,11,13,14,17,19,21,23,26,29,31,35,37,38,39,41,43,46,47,49,
%T A331784 53,57,58,59,61,65,67,69,71,73,74,77,79,83,87,89,91,94,95,97,98,101,
%U A331784 103,106,107,109,111,113,115,119,122,127,131,133,137,139,141,142
%N A331784 Lexicographically earliest sequence of positive integers that have at most one prime index already in the sequence, counting multiplicity.
%C A331784 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 A331784 Conjecture: A331912(n)/a(n) -> 1 as n -> infinity.
%H A331784 Gus Wiseman, <a href="/A331784/a331784.png">Plot of A331912(n)/A331784(n) for n = 1..3729.</a>
%e A331784 The sequence of terms together with their prime indices begins:
%e A331784     1: {}        43: {14}       91: {4,6}      141: {2,15}
%e A331784     2: {1}       46: {1,9}      94: {1,15}     142: {1,20}
%e A331784     3: {2}       47: {15}       95: {3,8}      143: {5,6}
%e A331784     5: {3}       49: {4,4}      97: {25}       145: {3,10}
%e A331784     7: {4}       53: {16}       98: {1,4,4}    147: {2,4,4}
%e A331784    11: {5}       57: {2,8}     101: {26}       149: {35}
%e A331784    13: {6}       58: {1,10}    103: {27}       151: {36}
%e A331784    14: {1,4}     59: {17}      106: {1,16}     157: {37}
%e A331784    17: {7}       61: {18}      107: {28}       158: {1,22}
%e A331784    19: {8}       65: {3,6}     109: {29}       159: {2,16}
%e A331784    21: {2,4}     67: {19}      111: {2,12}     161: {4,9}
%e A331784    23: {9}       69: {2,9}     113: {30}       163: {38}
%e A331784    26: {1,6}     71: {20}      115: {3,9}      167: {39}
%e A331784    29: {10}      73: {21}      119: {4,7}      169: {6,6}
%e A331784    31: {11}      74: {1,12}    122: {1,18}     173: {40}
%e A331784    35: {3,4}     77: {4,5}     127: {31}       178: {1,24}
%e A331784    37: {12}      79: {22}      131: {32}       179: {41}
%e A331784    38: {1,8}     83: {23}      133: {4,8}      181: {42}
%e A331784    39: {2,6}     87: {2,10}    137: {33}       182: {1,4,6}
%e A331784    41: {13}      89: {24}      139: {34}       183: {2,18}
%e A331784 For example, the prime indices of 95 are {3,8}, of which only 3 is in the sequence, so 95 is in the sequence.
%t A331784 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A331784 aQ[n_]:=Length[Cases[primeMS[n],_?aQ]]<=1;
%t A331784 Select[Range[100],aQ]
%Y A331784 Contains all prime numbers A000040.
%Y A331784 Numbers S without all prime indices in S are A324694.
%Y A331784 Numbers S without any prime indices in S are A324695.
%Y A331784 Numbers S with exactly one prime index in S are A331785.
%Y A331784 Numbers S with at most one distinct prime index in S are A331912.
%Y A331784 Numbers S with exactly one distinct prime index in S are A331913.
%Y A331784 Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331914.
%K A331784 nonn
%O A331784 1,2
%A A331784 _Gus Wiseman_, Feb 01 2020