%I #14 Mar 20 2019 04:22:25

%S 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 107,112,118,119,128,131,133,134,136,139,152,163,172,191,196,212,214,

%U 224,227,236,238,241,256,262,263,266,268,272,277,278,289,301,304

%N Lexicographically earliest sequence containing 1 and all positive integers > 2 whose prime indices already belong to 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 A multiplicative semigroup: if x and y are in the sequence then so is x*y. - _Robert Israel_, Mar 19 2019

%H Robert Israel, <a href="/A324854/b324854.txt">Table of n, a(n) for n = 1..10000</a>

%H Gus Wiseman, <a href="/A324854/a324854_1.png">The rooted trees whose Matula-Goebel numbers are the first 64 terms</a>.

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

%e 1: {}

%e 4: {1,1}

%e 7: {4}

%e 8: {1,1,1}

%e 14: {1,4}

%e 16: {1,1,1,1}

%e 17: {7}

%e 19: {8}

%e 28: {1,1,4}

%e 32: {1,1,1,1,1}

%e 34: {1,7}

%e 38: {1,8}

%e 43: {14}

%e 49: {4,4}

%e 53: {16}

%e 56: {1,1,1,4}

%e 59: {17}

%e 64: {1,1,1,1,1,1}

%e 67: {19}

%e 68: {1,1,7}

%p S:= {1}:

%p for n from 3 to 400 do

%p if map(numtheory:-pi, numtheory:-factorset(n)) subset S then

%p S:= S union {n}

%p fi

%p od:

%p sort(convert(S,list)); # _Robert Israel_, Mar 19 2019

%t aQ[n_]:=Switch[n,1,True,2,False,_,And@@Cases[FactorInteger[n],{p_,k_}:>aQ[PrimePi[p]]]];

%t Select[Range[100],aQ]

%Y Cf. A000002, A000720, A001462, A079254, A112798, A276625, A290822, A304360.

%Y Cf. A324697, A324698, A324736, A324748, A324753, A324843, A324850, A324855.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 18 2019