This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331935 #4 Feb 03 2020 22:18:37
%S A331935 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,
%T A331935 56,57,63,64,69,72,74,76,78,81,84,86,91,92,96,98,104,106,108,111,112,
%U A331935 114,117,122,126,128,129,133,138,144,146,147,148,152,156,159
%N A331935 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees.
%C A331935 A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.
%C A331935 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.
%C A331935 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.
%H A331935 David Callan, <a href="http://arxiv.org/abs/1406.7784">A sign-reversing involution to count labeled lone-child-avoiding trees</a>, arXiv:1406.7784 [math.CO], (30-June-2014).
%H A331935 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vS1zCO9fgAIe5rGiAhTtlrOTuqsmuPos2zkeFPYB80gNzLb44ufqIqksTB4uM9SIpwlvo-oOHhepywy/pub">Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.</a>
%e A331935 The sequence of all semi-lone-child-avoiding rooted trees together with their Matula-Goebel numbers begins:
%e A331935 1: o
%e A331935 2: (o)
%e A331935 4: (oo)
%e A331935 6: (o(o))
%e A331935 8: (ooo)
%e A331935 9: ((o)(o))
%e A331935 12: (oo(o))
%e A331935 14: (o(oo))
%e A331935 16: (oooo)
%e A331935 18: (o(o)(o))
%e A331935 21: ((o)(oo))
%e A331935 24: (ooo(o))
%e A331935 26: (o(o(o)))
%e A331935 27: ((o)(o)(o))
%e A331935 28: (oo(oo))
%e A331935 32: (ooooo)
%e A331935 36: (oo(o)(o))
%e A331935 38: (o(ooo))
%e A331935 39: ((o)(o(o)))
%e A331935 42: (o(o)(oo))
%e A331935 The sequence of terms together with their prime indices begins:
%e A331935 1: {} 46: {1,9} 98: {1,4,4}
%e A331935 2: {1} 48: {1,1,1,1,2} 104: {1,1,1,6}
%e A331935 4: {1,1} 49: {4,4} 106: {1,16}
%e A331935 6: {1,2} 52: {1,1,6} 108: {1,1,2,2,2}
%e A331935 8: {1,1,1} 54: {1,2,2,2} 111: {2,12}
%e A331935 9: {2,2} 56: {1,1,1,4} 112: {1,1,1,1,4}
%e A331935 12: {1,1,2} 57: {2,8} 114: {1,2,8}
%e A331935 14: {1,4} 63: {2,2,4} 117: {2,2,6}
%e A331935 16: {1,1,1,1} 64: {1,1,1,1,1,1} 122: {1,18}
%e A331935 18: {1,2,2} 69: {2,9} 126: {1,2,2,4}
%e A331935 21: {2,4} 72: {1,1,1,2,2} 128: {1,1,1,1,1,1,1}
%e A331935 24: {1,1,1,2} 74: {1,12} 129: {2,14}
%e A331935 26: {1,6} 76: {1,1,8} 133: {4,8}
%e A331935 27: {2,2,2} 78: {1,2,6} 138: {1,2,9}
%e A331935 28: {1,1,4} 81: {2,2,2,2} 144: {1,1,1,1,2,2}
%e A331935 32: {1,1,1,1,1} 84: {1,1,2,4} 146: {1,21}
%e A331935 36: {1,1,2,2} 86: {1,14} 147: {2,4,4}
%e A331935 38: {1,8} 91: {4,6} 148: {1,1,12}
%e A331935 39: {2,6} 92: {1,1,9} 152: {1,1,1,8}
%e A331935 42: {1,2,4} 96: {1,1,1,1,1,2} 156: {1,1,2,6}
%t A331935 mseQ[n_]:=n==1||n==2||!PrimeQ[n]&&And@@mseQ/@PrimePi/@First/@FactorInteger[n];
%t A331935 Select[Range[100],mseQ]
%Y A331935 The enumeration of these trees by leaves is A050381.
%Y A331935 The locally disjoint version A331873.
%Y A331935 The enumeration of these trees by nodes is A331934.
%Y A331935 The case with at most one distinct non-leaf branch of any vertex is A331936.
%Y A331935 Lone-child-avoiding rooted trees are counted by A001678.
%Y A331935 Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
%Y A331935 Cf. A000081, A000669, A007097, A061775, A196050, A320269, A330951, A331871, A331872, A331874, A331937, A331965.
%K A331935 nonn
%O A331935 1,2
%A A331935 _Gus Wiseman_, Feb 03 2020