%I #23 Jun 10 2020 14:06:10

%S 1,2,4,6,8,12,14,16,24,26,28,32,38,48,52,56,64,74,76,86,96,104,106,

%T 112,128,148,152,172,178,192,202,208,212,214,224,256,262,296,304,326,

%U 344,356,384,404,416,424,428,446,448,478,512,524,526,592,608,622,652

%N One, two, and all numbers of the form 2^k * prime(j) where k > 0 and j already belongs to the sequence.

%C Also Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted semi-identity trees. A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct. Note that these conditions together imply that there is at most one non-leaf branch under any given vertex.

%C Also Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex.

%C The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches (of the root), which gives a bijective correspondence between positive integers and unlabeled rooted trees.

%H Robert Israel, <a href="/A331681/b331681.txt">Table of n, a(n) for n = 1..4000</a>

%F Intersection of A306202 (semi-identity), A316495 (locally disjoint), and A331935 (semi-lone-child-avoiding). - _Gus Wiseman_, Jun 09 2020

%e The sequence of all semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex, together with their Matula-Goebel numbers, begins:

%e 1: o

%e 2: (o)

%e 4: (oo)

%e 6: (o(o))

%e 8: (ooo)

%e 12: (oo(o))

%e 14: (o(oo))

%e 16: (oooo)

%e 24: (ooo(o))

%e 26: (o(o(o)))

%e 28: (oo(oo))

%e 32: (ooooo)

%e 38: (o(ooo))

%e 48: (oooo(o))

%e 52: (oo(o(o)))

%e 56: (ooo(oo))

%e 64: (oooooo)

%e 74: (o(oo(o)))

%e 76: (oo(ooo))

%e 86: (o(o(oo)))

%p N:= 1000: # for terms <= N

%p S:= {1,2}:

%p with(queue):

%p Q:= new(1,2):

%p while not empty(Q) do

%p r:= dequeue(Q);

%p p:= ithprime(r);

%p newS:= {seq(2^i*p,i=1..ilog2(N/p))} minus S;

%p S:= S union newS;

%p for s in newS do enqueue(Q,s) od:

%p od:

%p sort(convert(S,list)); # _Robert Israel_, Feb 05 2020

%t uryQ[n_]:=n==1||MatchQ[FactorInteger[n],({{2,_},{p_,1}}/;uryQ[PrimePi[p]])|{{2,_}}];

%t Select[Range[100],uryQ]

%Y The enumeration of these trees by nodes is A324969 (essentially A000045).

%Y The enumeration of these trees by leaves appears to be A090129(n + 1).

%Y The (non-semi) lone-child-avoiding version is A331683.

%Y Matula-Goebel numbers of rooted semi-identity trees are A306202.

%Y Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.

%Y The set S of numbers with at most one prime index in S is A331784.

%Y Matula-Goebel numbers of locally disjoint rooted trees are A316495.

%Y Cf. A000081, A000669, A001678, A007097, A061775, A196050, A291636, A316470, A316473, A331679, A331680, A331682, A331687, A331783, A331785, A331873.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 26 2020