A297571 Matula-Goebel numbers of fully unbalanced 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

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.

Links

Table of n, a(n) for n=1..59.

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]

Crossrefs

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

Sequence in context: A302534 A063451 A276625 * A316503 A316494 A178992

Adjacent sequences: A297568 A297569 A297570 * A297572 A297573 A297574

Keyword

nonn

Author

Gus Wiseman, Dec 31 2017

Status

approved