A306203 Matula-Goebel numbers of balanced rooted semi-identity trees.

1, 2, 3, 4, 5, 7, 8, 11, 16, 17, 19, 21, 31, 32, 53, 57, 59, 64, 67, 73, 85, 127, 128, 131, 133, 159, 241, 256, 269, 277, 311, 331, 335, 365, 367, 371, 393, 399, 439, 512, 649, 709, 719, 739, 751, 917, 933, 937, 1007, 1024, 1113, 1139, 1205, 1241, 1345, 1523

Offset

1,2

Comments

A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. It is balanced if all leaves are the same distance from the root. The only balanced rooted identity trees are rooted paths.

Links

Table of n, a(n) for n=1..56.

Example

The sequence of all unlabeled balanced rooted semi-identity trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  21: ((o)(oo))
  31: (((((o)))))
  32: (ooooo)
  53: ((oooo))
  57: ((o)(ooo))
  59: ((((oo))))
  64: (oooooo)
  67: (((ooo)))
  73: (((o)(oo)))
  85: (((o))((oo)))

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
psidQ[n_]:=And[UnsameQ@@DeleteCases[primeMS[n], 1], And@@psidQ/@primeMS[n]];
mgtree[n_]:=If[n==1, {}, mgtree/@primeMS[n]];
Select[Range[100], And[psidQ[#], SameQ@@Length/@Position[mgtree[#], {}]]&]

Crossrefs

Cf. A000081, A004111, A007097, A048816, A184155, A276625, A306200, A306201, A306202, A316467, A317710.

Sequence in context: A241340 A326667 A293441 * A111795 A046098 A258085

Adjacent sequences: A306200 A306201 A306202 * A306204 A306205 A306206

Keyword

nonn

Author

Gus Wiseman, Jan 29 2019

Status

approved