A331935 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees.

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 21, 24, 26, 27, 28, 32, 36, 38, 39, 42, 46, 48, 49, 52, 54, 56, 57, 63, 64, 69, 72, 74, 76, 78, 81, 84, 86, 91, 92, 96, 98, 104, 106, 108, 111, 112, 114, 117, 122, 126, 128, 129, 133, 138, 144, 146, 147, 148, 152, 156, 159

Offset: 1

Author

Gus Wiseman, Feb 03 2020

Keywords

• nonn

Comments

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the 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 one, two, and all nonprime numbers whose prime indices already belong to the sequence, where a prime index of n is a number m such that prime(m) divides n.

Examples

			The sequence of all semi-lone-child-avoiding rooted trees 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))
  21: ((o)(oo))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  39: ((o)(o(o)))
  42: (o(o)(oo))
The sequence of terms together with their prime indices begins:
    1: {}              46: {1,9}             98: {1,4,4}
    2: {1}             48: {1,1,1,1,2}      104: {1,1,1,6}
    4: {1,1}           49: {4,4}            106: {1,16}
    6: {1,2}           52: {1,1,6}          108: {1,1,2,2,2}
    8: {1,1,1}         54: {1,2,2,2}        111: {2,12}
    9: {2,2}           56: {1,1,1,4}        112: {1,1,1,1,4}
   12: {1,1,2}         57: {2,8}            114: {1,2,8}
   14: {1,4}           63: {2,2,4}          117: {2,2,6}
   16: {1,1,1,1}       64: {1,1,1,1,1,1}    122: {1,18}
   18: {1,2,2}         69: {2,9}            126: {1,2,2,4}
   21: {2,4}           72: {1,1,1,2,2}      128: {1,1,1,1,1,1,1}
   24: {1,1,1,2}       74: {1,12}           129: {2,14}
   26: {1,6}           76: {1,1,8}          133: {4,8}
   27: {2,2,2}         78: {1,2,6}          138: {1,2,9}
   28: {1,1,4}         81: {2,2,2,2}        144: {1,1,1,1,2,2}
   32: {1,1,1,1,1}     84: {1,1,2,4}        146: {1,21}
   36: {1,1,2,2}       86: {1,14}           147: {2,4,4}
   38: {1,8}           91: {4,6}            148: {1,1,12}
   39: {2,6}           92: {1,1,9}          152: {1,1,1,8}
   42: {1,2,4}         96: {1,1,1,1,1,2}    156: {1,1,2,6}
		

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

The enumeration of these trees by leaves is A050381.

The locally disjoint version A331873.

The enumeration of these trees by nodes is A331934.

The case with at most one distinct non-leaf branch of any vertex is A331936.

Lone-child-avoiding rooted trees are counted by A001678.

Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.

Cf. A000081, A000669, A007097, A061775, A196050, A320269, A330951, A331871, A331872, A331874, A331937, A331965.

Programs

Mathematica

mseQ[n_]:=n==1||n==2||!PrimeQ[n]&&And@@mseQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],mseQ]