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A324854 Lexicographically earliest sequence containing 1 and all positive integers > 2 whose prime indices already belong to the sequence.

%I A324854 #14 Mar 20 2019 04:22:25
%S A324854 1,4,7,8,14,16,17,19,28,32,34,38,43,49,53,56,59,64,67,68,76,86,98,106,
%T A324854 107,112,118,119,128,131,133,134,136,139,152,163,172,191,196,212,214,
%U A324854 224,227,236,238,241,256,262,263,266,268,272,277,278,289,301,304
%N A324854 Lexicographically earliest sequence containing 1 and all positive integers > 2 whose prime indices already belong to the sequence.
%C A324854 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 A324854 A multiplicative semigroup: if x and y are in the sequence then so is x*y. - _Robert Israel_, Mar 19 2019
%H A324854 Robert Israel, <a href="/A324854/b324854.txt">Table of n, a(n) for n = 1..10000</a>
%H A324854 Gus Wiseman, <a href="/A324854/a324854_1.png">The rooted trees whose Matula-Goebel numbers are the first 64 terms</a>.
%e A324854 The sequence of terms together with their prime indices begins:
%e A324854    1: {}
%e A324854    4: {1,1}
%e A324854    7: {4}
%e A324854    8: {1,1,1}
%e A324854   14: {1,4}
%e A324854   16: {1,1,1,1}
%e A324854   17: {7}
%e A324854   19: {8}
%e A324854   28: {1,1,4}
%e A324854   32: {1,1,1,1,1}
%e A324854   34: {1,7}
%e A324854   38: {1,8}
%e A324854   43: {14}
%e A324854   49: {4,4}
%e A324854   53: {16}
%e A324854   56: {1,1,1,4}
%e A324854   59: {17}
%e A324854   64: {1,1,1,1,1,1}
%e A324854   67: {19}
%e A324854   68: {1,1,7}
%p A324854 S:= {1}:
%p A324854 for n from 3 to 400 do
%p A324854   if map(numtheory:-pi, numtheory:-factorset(n)) subset S then
%p A324854     S:= S union {n}
%p A324854   fi
%p A324854 od:
%p A324854 sort(convert(S,list)); # _Robert Israel_, Mar 19 2019
%t A324854 aQ[n_]:=Switch[n,1,True,2,False,_,And@@Cases[FactorInteger[n],{p_,k_}:>aQ[PrimePi[p]]]];
%t A324854 Select[Range[100],aQ]
%Y A324854 Cf. A000002, A000720, A001462, A079254, A112798, A276625, A290822, A304360.
%Y A324854 Cf. A324697, A324698, A324736, A324748, A324753, A324843, A324850, A324855.
%K A324854 nonn
%O A324854 1,2
%A A324854 _Gus Wiseman_, Mar 18 2019