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

Author

Gus Wiseman, Jan 29 2019

Keywords

• nonn

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.

Examples

			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)))
		

Crossrefs

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

Programs

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[#],{}]]&]