A320269 Matula-Goebel numbers of lone-child-avoiding rooted trees in which the non-leaf branches directly under any given node are all equal (semi-achirality).

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052

Offset: 1

Author

Gus Wiseman, Oct 08 2018

Keywords

• nonn

Comments

First differs from A331871 in lacking 1589.

Lone-child-avoiding means there are no unary branchings.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Examples

			The sequence of rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))
		

Links

• Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Crossrefs

Cf. A002541, A070776, A167865, A214577, A317710, A320222, A320226.

The same-tree version is A291441.

Not requiring lone-child-avoidance gives A320230.

The enumeration of these trees by vertices is A320268.

The semi-lone-child-avoiding version is A331936.

If the non-leaf branches are all different instead of equal we get A331965.

The fully-achiral case is A331967.

Achiral rooted trees are counted by A003238.

MG-numbers of lone-child-avoiding rooted trees are A291636.

Cf. A061775, A196050, A276625, A280996, A306202, A331912, A331916, A331966.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
hmakQ[n_]:=And[!PrimeQ[n],SameQ@@DeleteCases[primeMS[n],1],And@@hmakQ/@primeMS[n]];Select[Range[1000],hmakQ[#]&]

Extensions

Updated with corrected terminology by Gus Wiseman, Feb 06 2020