OFFSET
1,2
COMMENTS
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.
To get a permutation of a tree, we choose a permutation of the multiset of branches of each node.
EXAMPLE
The terms together with their corresponding trees begin:
1: o
6: (o(o))
12: (oo(o))
24: (ooo(o))
48: (oooo(o))
30: (o(o)((o)))
192: (oooooo(o))
104: (ooo(o(o)))
148: (oo(oo(o)))
72: (ooo(o)(o))
3072: (oooooooooo(o))
60: (oo(o)((o)))
12288: (oooooooooooo(o))
832: (oooooo(o(o)))
144: (oooo(o)(o))
712: (ooo(ooo(o)))
MATHEMATICA
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]
MGTree[n_Integer]:=If[n===1, {}, MGTree/@primeMS[n]]
treeperms[t_]:=Times @@ Cases[t, b:{__}:>Length[Permutations[b]], {0, Infinity}];
uv=Table[treeperms[MGTree[n]], {n, 100000}];
Table[Position[uv, k][[1, 1]], {k, Min@@Complement[Range[Max@@uv], uv]-1}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 20 2022
STATUS
approved