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A324769
Matula-Goebel numbers of fully anti-transitive rooted trees.
5
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 64, 65, 67, 71, 73, 77, 79, 81, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 129, 131, 133, 137, 139, 143, 147
OFFSET
1,2
COMMENTS
An unlabeled rooted tree is fully anti-transitive if no proper terminal subtree of any branch of the root is a branch of the root.
EXAMPLE
The sequence of fully anti-transitive rooted trees together with their Matula-Goebel numbers begins:
1: o
2: (o)
3: ((o))
4: (oo)
5: (((o)))
7: ((oo))
8: (ooo)
9: ((o)(o))
11: ((((o))))
13: ((o(o)))
16: (oooo)
17: (((oo)))
19: ((ooo))
21: ((o)(oo))
23: (((o)(o)))
25: (((o))((o)))
27: ((o)(o)(o))
29: ((o((o))))
31: (((((o)))))
32: (ooooo)
35: (((o))(oo))
37: ((oo(o)))
41: (((o(o))))
43: ((o(oo)))
47: (((o)((o))))
49: ((oo)(oo))
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
fullantiQ[n_]:=Intersection[Union@@Rest[FixedPointList[Union@@primeMS/@#&, primeMS[n]]], primeMS[n]]=={};
Select[Range[100], fullantiQ]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 17 2019
STATUS
approved