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A331871
Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.
9
1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052
OFFSET
1,2
COMMENTS
First differs from A320269 in having 1589, the Matula-Goebel number of the tree ((oo)((oo)(oo))).
First differs from A331683 in having 49.
A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.
Lone-child-avoiding means 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 its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Consists of one and all nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.
FORMULA
Intersection of A291636 and A316495.
EXAMPLE
The sequence of all lone-child-avoiding locally disjoint rooted trees 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))
49: ((oo)(oo))
56: (ooo(oo))
64: (oooooo)
76: (oo(ooo))
86: (o(o(oo)))
98: (o(oo)(oo))
106: (o(oooo))
112: (oooo(oo))
128: (ooooooo)
152: (ooo(ooo))
172: (oo(o(oo)))
196: (oo(oo)(oo))
The sequence of terms together with their prime indices begins:
1: {} 212: {1,1,16}
4: {1,1} 214: {1,28}
8: {1,1,1} 224: {1,1,1,1,1,4}
14: {1,4} 256: {1,1,1,1,1,1,1,1}
16: {1,1,1,1} 262: {1,32}
28: {1,1,4} 304: {1,1,1,1,8}
32: {1,1,1,1,1} 326: {1,38}
38: {1,8} 343: {4,4,4}
49: {4,4} 344: {1,1,1,14}
56: {1,1,1,4} 361: {8,8}
64: {1,1,1,1,1,1} 392: {1,1,1,4,4}
76: {1,1,8} 424: {1,1,1,16}
86: {1,14} 428: {1,1,28}
98: {1,4,4} 448: {1,1,1,1,1,1,4}
106: {1,16} 454: {1,49}
112: {1,1,1,1,4} 512: {1,1,1,1,1,1,1,1,1}
128: {1,1,1,1,1,1,1} 524: {1,1,32}
152: {1,1,1,8} 526: {1,56}
172: {1,1,14} 608: {1,1,1,1,1,8}
196: {1,1,4,4} 622: {1,64}
MATHEMATICA
msQ[n_]:=n==1||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[1000], msQ]
CROSSREFS
Not requiring local disjointness gives A291636.
Not requiring lone-child avoidance gives A316495.
A superset of A320269.
These trees are counted by A331680.
The semi-identity tree version is A331683.
The version containing 2 is A331873.
Sequence in context: A312337 A291636 A320269 * A331965 A331683 A036312
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 02 2020
STATUS
approved