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A324855 Lexicographically earliest sequence containing 2 and all squarefree numbers > 2 whose prime indices already belong to the sequence.

%I A324855 #10 Mar 23 2019 02:37:19
%S A324855 2,3,5,11,15,31,33,47,55,93,127,137,141,155,165,211,235,257,341,381,
%T A324855 411,465,487,517,633,635,685,705,709,771,773,811,907,977,1023,1055,
%U A324855 1285,1297,1397,1457,1461,1483,1507,1551,1621,1705,1905,2055,2127,2293,2319
%N A324855 Lexicographically earliest sequence containing 2 and all squarefree numbers > 2 whose prime indices already belong to the sequence.
%C A324855 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.
%H A324855 Robert Israel, <a href="/A324855/b324855.txt">Table of n, a(n) for n = 1..1567</a>
%H A324855 Gus Wiseman, <a href="/A324855/a324855_1.png">The rooted identity trees whose Matula-Goebel numbers are the first 64 terms</a>.
%e A324855 The sequence of terms together with their prime indices begins:
%e A324855     2: {1}
%e A324855     3: {2}
%e A324855     5: {3}
%e A324855    11: {5}
%e A324855    15: {2,3}
%e A324855    31: {11}
%e A324855    33: {2,5}
%e A324855    47: {15}
%e A324855    55: {3,5}
%e A324855    93: {2,11}
%e A324855   127: {31}
%e A324855   137: {33}
%e A324855   141: {2,15}
%e A324855   155: {3,11}
%e A324855   165: {2,3,5}
%e A324855   211: {47}
%e A324855   235: {3,15}
%e A324855   257: {55}
%e A324855   341: {5,11}
%e A324855   381: {2,31}
%p A324855 S:= {2}: count:= 1:
%p A324855 for n from 3 by 2 while count < 100 do
%p A324855   F:= ifactors(n)[2];
%p A324855   if max(map(t -> t[2],F))=1 and {seq(numtheory:-pi(t[1]),t=F)} subset S then
%p A324855      S:= S union {n}; count:= count+1;
%p A324855   fi
%p A324855 od:
%p A324855 sort(convert(S,list)); # _Robert Israel_, Mar 22 2019
%t A324855 aQ[n_]:=Switch[n,1,False,2,True,_?(!SquareFreeQ[#]&),False,_,And@@Cases[FactorInteger[n],{p_,k_}:>aQ[PrimePi[p]]]];
%t A324855 Select[Range[1000],aQ]
%Y A324855 Cf. A000002, A000720, A001462, A079254, A109298, A112798, A276625, A290822.
%Y A324855 Cf. A324697, A324698, A324736, A324748, A324753, A324843, A324850, A324854.
%Y A324855 Contains A007097 except for 1.
%K A324855 nonn
%O A324855 1,1
%A A324855 _Gus Wiseman_, Mar 18 2019