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A331681
One, two, and all numbers of the form 2^k * prime(j) where k > 0 and j already belongs to the sequence.
12
1, 2, 4, 6, 8, 12, 14, 16, 24, 26, 28, 32, 38, 48, 52, 56, 64, 74, 76, 86, 96, 104, 106, 112, 128, 148, 152, 172, 178, 192, 202, 208, 212, 214, 224, 256, 262, 296, 304, 326, 344, 356, 384, 404, 416, 424, 428, 446, 448, 478, 512, 524, 526, 592, 608, 622, 652
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
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 nodes is A324969 (essentially A000045).
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.
Sequence in context: A331994 A043723 A376508 * A006998 A043726 A043730
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 26 2020
STATUS
approved