OFFSET
1,2
COMMENTS
Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf.
In a semi-identity tree, the non-leaf branches of any given vertex are distinct.
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 numbers that can be written as a power of two (other than 2) times a squarefree number whose prime indices already belong to the sequence, where a prime index of n is a number m such that prime(m) divides n.
LINKS
EXAMPLE
The sequence of all semi-lone-child-avoiding rooted semi-identity trees together with their Matula-Goebel numbers begins:
1: o
2: (o)
4: (oo)
6: (o(o))
8: (ooo)
12: (oo(o))
14: (o(oo))
16: (oooo)
21: ((o)(oo))
24: (ooo(o))
26: (o(o(o)))
28: (oo(oo))
32: (ooooo)
38: (o(ooo))
39: ((o)(o(o)))
42: (o(o)(oo))
48: (oooo(o))
52: (oo(o(o)))
56: (ooo(oo))
57: ((o)(ooo))
The sequence of terms together with their prime indices begins:
1: {} 64: {1,1,1,1,1,1} 159: {2,16}
2: {1} 74: {1,12} 168: {1,1,1,2,4}
4: {1,1} 76: {1,1,8} 172: {1,1,14}
6: {1,2} 78: {1,2,6} 178: {1,24}
8: {1,1,1} 84: {1,1,2,4} 182: {1,4,6}
12: {1,1,2} 86: {1,14} 192: {1,1,1,1,1,1,2}
14: {1,4} 91: {4,6} 202: {1,26}
16: {1,1,1,1} 96: {1,1,1,1,1,2} 208: {1,1,1,1,6}
21: {2,4} 104: {1,1,1,6} 212: {1,1,16}
24: {1,1,1,2} 106: {1,16} 214: {1,28}
26: {1,6} 111: {2,12} 219: {2,21}
28: {1,1,4} 112: {1,1,1,1,4} 222: {1,2,12}
32: {1,1,1,1,1} 114: {1,2,8} 224: {1,1,1,1,1,4}
38: {1,8} 128: {1,1,1,1,1,1,1} 228: {1,1,2,8}
39: {2,6} 129: {2,14} 247: {6,8}
42: {1,2,4} 133: {4,8} 256: {1,1,1,1,1,1,1,1}
48: {1,1,1,1,2} 146: {1,21} 258: {1,2,14}
52: {1,1,6} 148: {1,1,12} 259: {4,12}
56: {1,1,1,4} 152: {1,1,1,8} 262: {1,32}
57: {2,8} 156: {1,1,2,6} 266: {1,4,8}
MATHEMATICA
scsiQ[n_]:=n==1||n==2||!PrimeQ[n]&&FreeQ[FactorInteger[n], {_?(#>2&), _?(#>1&)}]&&And@@scsiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100], scsiQ]
CROSSREFS
The locally disjoint version is A331681.
The enumeration of these trees by vertices is A331993.
Semi-identity trees are A306200.
MG-numbers of rooted identity trees are A276625.
MG-numbers of lone-child-avoiding rooted identity trees are {1}.
MG-numbers of lone-child-avoiding rooted trees are A291636.
MG-numbers of semi-identity trees are A306202.
MG-numbers of semi-lone-child-avoiding rooted trees are A331935.
MG-numbers of semi-lone-child-avoiding rooted identity trees are A331963.
MG-numbers of lone-child-avoiding rooted semi-identity trees are A331965.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 05 2020
STATUS
approved