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EXAMPLE
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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}
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