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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))))

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proposed

Matula-Goebel numbers of fully imbalanced unbalanced rooted trees.

An unlabeled rooted tree is fully imbalanced 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 imbalanced unbalanced also. The number of fully imbalanced unbalanced trees with n nodes is A032305(n).

Sequence of fully imbalanced unbalanced trees begins:

proposed

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proposed

allocated for Gus WisemanMatula-Goebel numbers of fully imbalanced rooted trees.

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

1,2

An unlabeled rooted tree is fully imbalanced if either (1) it is a single node, or (2a) every branch has a different number of nodes and (2b) every branch is fully imbalanced also. The number of fully imbalanced trees with n nodes is A032305(n).

The first finitary number (A276625) not in this sequence is 143.

Sequence of fully imbalanced 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))))

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]

Cf. A000081, A003238, A004111, A007097, A032305, A061775, A214577, A273873, A276625, A277098, A290689, A290760, A291441, A291442, A291443.

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Gus Wiseman, Dec 31 2017

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editing

allocated for Gus Wiseman

allocated

approved