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A331683
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One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.
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20
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1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 152, 172, 212, 214, 224, 256, 262, 304, 326, 344, 424, 428, 448, 512, 524, 526, 608, 622, 652, 688, 766, 848, 856, 886, 896, 1024, 1048, 1052, 1154, 1216, 1226, 1244, 1304, 1376, 1438, 1532, 1696
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OFFSET
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1,2
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COMMENTS
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Also Matula-Goebel numbers of lone-child-avoiding rooted trees at with at most one non-leaf branch under any given vertex. A rooted tree is lone-child-avoiding if there are no unary branchings. The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of the root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Also Matula-Goebel numbers of lone-child-avoiding locally disjoint semi-identity trees. 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.
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LINKS
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FORMULA
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EXAMPLE
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The sequence of all 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
4: (oo)
8: (ooo)
14: (o(oo))
16: (oooo)
28: (oo(oo))
32: (ooooo)
38: (o(ooo))
56: (ooo(oo))
64: (oooooo)
76: (oo(ooo))
86: (o(o(oo)))
106: (o(oooo))
112: (oooo(oo))
128: (ooooooo)
152: (ooo(ooo))
172: (oo(o(oo)))
212: (oo(oooo))
214: (o(oo(oo)))
224: (ooooo(oo))
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MAPLE
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N:= 10^4: # for terms <= N
S:= {1}:
with(queue):
Q:= new(1):
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:
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MATHEMATICA
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uryQ[n_]:=n==1||MatchQ[FactorInteger[n], ({{2, _}, {p_, 1}}/; uryQ[PrimePi[p]])|({{2, k_}}/; k>1)];
Select[Range[100], uryQ]
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CROSSREFS
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These trees counted by number of vertices are A212804.
The semi-lone-child-avoiding version is A331681.
The non-semi-identity version is A331871.
Lone-child-avoiding rooted trees are counted by A001678.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Unlabeled semi-identity trees are counted by A306200, with Matula-Goebel numbers A306202.
Locally disjoint rooted trees are counted by A316473.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.
Cf. A000081, A007097, A061775, A196050, A276625, A316470, A316696, A331678, A331679, A331680, A331682, A331686, A331687.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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