OFFSET
1,2
COMMENTS
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.
Also Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex.
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.
LINKS
Robert Israel, Table of n, a(n) for n = 1..4000
FORMULA
Intersection of A306202 (semi-identity), A316495 (locally disjoint), and A331935 (semi-lone-child-avoiding). - Gus Wiseman, Jun 09 2020
EXAMPLE
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:
1: o
2: (o)
4: (oo)
6: (o(o))
8: (ooo)
12: (oo(o))
14: (o(oo))
16: (oooo)
24: (ooo(o))
26: (o(o(o)))
28: (oo(oo))
32: (ooooo)
38: (o(ooo))
48: (oooo(o))
52: (oo(o(o)))
56: (ooo(oo))
64: (oooooo)
74: (o(oo(o)))
76: (oo(ooo))
86: (o(o(oo)))
MAPLE
N:= 1000: # for terms <= N
S:= {1, 2}:
with(queue):
Q:= new(1, 2):
while not empty(Q) do
r:= dequeue(Q);
p:= ithprime(r);
newS:= {seq(2^i*p, i=1..ilog2(N/p))} minus S;
S:= S union newS;
for s in newS do enqueue(Q, s) od:
od:
sort(convert(S, list)); # Robert Israel, Feb 05 2020
MATHEMATICA
uryQ[n_]:=n==1||MatchQ[FactorInteger[n], ({{2, _}, {p_, 1}}/; uryQ[PrimePi[p]])|{{2, _}}];
Select[Range[100], uryQ]
CROSSREFS
The enumeration of these trees by leaves appears to be A090129(n + 1).
The (non-semi) lone-child-avoiding version is A331683.
Matula-Goebel numbers of rooted semi-identity trees are A306202.
Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.
The set S of numbers with at most one prime index in S is A331784.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 26 2020
STATUS
approved