A331936 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 144, 148, 152, 162, 169, 172, 178, 184, 192, 196, 202, 206, 208, 212, 214, 216, 224, 243, 244, 256, 262, 288

Offset: 1

Author

Gus Wiseman, Feb 03 2020

Keywords

• nonn

Comments

First differs from A331873 in lacking 69, the Matula-Goebel number of the tree ((o)((o)(o))).

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf.

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.

Consists of 1, 2, and all numbers equal to a power of 2 (other than 1) times a power of prime(j) for some j > 1 already in the sequence.

Examples

			The sequence of rooted trees ranked by this sequence together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  46: (o((o)(o)))
  48: (oooo(o))
  49: ((oo)(oo))
The sequence of terms together with their prime indices begins:
    1: {}              52: {1,1,6}            152: {1,1,1,8}
    2: {1}             54: {1,2,2,2}          162: {1,2,2,2,2}
    4: {1,1}           56: {1,1,1,4}          169: {6,6}
    6: {1,2}           64: {1,1,1,1,1,1}      172: {1,1,14}
    8: {1,1,1}         72: {1,1,1,2,2}        178: {1,24}
    9: {2,2}           74: {1,12}             184: {1,1,1,9}
   12: {1,1,2}         76: {1,1,8}            192: {1,1,1,1,1,1,2}
   14: {1,4}           81: {2,2,2,2}          196: {1,1,4,4}
   16: {1,1,1,1}       86: {1,14}             202: {1,26}
   18: {1,2,2}         92: {1,1,9}            206: {1,27}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      208: {1,1,1,1,6}
   26: {1,6}           98: {1,4,4}            212: {1,1,16}
   27: {2,2,2}        104: {1,1,1,6}          214: {1,28}
   28: {1,1,4}        106: {1,16}             216: {1,1,1,2,2,2}
   32: {1,1,1,1,1}    108: {1,1,2,2,2}        224: {1,1,1,1,1,4}
   36: {1,1,2,2}      112: {1,1,1,1,4}        243: {2,2,2,2,2}
   38: {1,8}          122: {1,18}             244: {1,1,18}
   46: {1,9}          128: {1,1,1,1,1,1,1}    256: {1,1,1,1,1,1,1,1}
   48: {1,1,1,1,2}    144: {1,1,1,1,2,2}      262: {1,32}
   49: {4,4}          148: {1,1,12}           288: {1,1,1,1,1,2,2}
		

Links

• David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
• Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Crossrefs

A superset of A000079.

The non-lone-child-avoiding version is A320230.

The non-semi version is A320269.

These trees are counted by A331933.

Not requiring semi-achirality gives A331935.

The fully-achiral case is A331992.

Achiral trees are counted by A003238.

Numbers with at most one distinct odd prime factor are A070776.

Matula-Goebel numbers of achiral rooted trees are A214577.

Matula-Goebel numbers of semi-identity trees are A306202.

Numbers S with at most one distinct prime index in S are A331912.

Cf. A001678, A007097, A050381, A061775, A196050, A291636, A331784, A331873, A331914, A331934, A331965, A331967, A331991.

Programs

Mathematica

msQ[n_]:=n<=2||!PrimeQ[n]&&Length[DeleteCases[FactorInteger[n],{2,_}]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],msQ]

Formula

Intersection of A320230 and A331935.