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A297571
Matula-Goebel numbers of fully unbalanced rooted trees.
7
1, 2, 3, 5, 6, 10, 11, 13, 15, 22, 26, 29, 30, 31, 33, 39, 41, 47, 55, 58, 62, 65, 66, 78, 79, 82, 87, 93, 94, 101, 109, 110, 113, 123, 127, 130, 137, 141, 145, 155, 158, 165, 167, 174, 179, 186, 195, 202, 205, 211, 218, 226, 235, 237, 246, 254, 257, 271, 274
OFFSET
1,2
COMMENTS
An unlabeled rooted tree is fully unbalanced if either (1) it is a single node, or (2a) every branch has a different number of nodes and (2b) every branch is fully unbalanced also. The number of fully unbalanced trees with n nodes is A032305(n).
The first finitary number (A276625) not in this sequence is 143.
EXAMPLE
Sequence of fully unbalanced trees begins:
1 o
2 (o)
3 ((o))
5 (((o)))
6 (o(o))
10 (o((o)))
11 ((((o))))
13 ((o(o)))
15 ((o)((o)))
22 (o(((o))))
26 (o(o(o)))
29 ((o((o))))
30 (o(o)((o)))
31 (((((o)))))
33 ((o)(((o))))
39 ((o)(o(o)))
41 (((o(o))))
47 (((o)((o))))
MATHEMATICA
nn=2000;
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
MGweight[n_]:=If[n===1, 1, 1+Total[Cases[FactorInteger[n], {p_, k_}:>k*MGweight[PrimePi[p]]]]];
imbalQ[n_]:=Or[n===1, With[{m=primeMS[n]}, And[UnsameQ@@MGweight/@m, And@@imbalQ/@m]]];
Select[Range[nn], imbalQ]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 31 2017
STATUS
approved