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 A331871 #9 Feb 03 2020 22:17:51
%S A331871 1,4,8,14,16,28,32,38,49,56,64,76,86,98,106,112,128,152,172,196,212,
%T A331871 214,224,256,262,304,326,343,344,361,392,424,428,448,454,512,524,526,
%U A331871 608,622,652,686,688,722,766,784,848,856,886,896,908,1024,1042,1048,1052
%N A331871 Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.
%C A331871 First differs from A320269 in having 1589, the Matula-Goebel number of the tree ((oo)((oo)(oo))).
%C A331871 First differs from A331683 in having 49.
%C A331871 A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.
%C A331871 Lone-child-avoiding means there are no unary branchings.
%C A331871 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 A331871 Consists of one and all nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.
%H A331871 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 A331871 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>
%F A331871 Intersection of A291636 and A316495.
%e A331871 The sequence of all lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
%e A331871 1: o
%e A331871 4: (oo)
%e A331871 8: (ooo)
%e A331871 14: (o(oo))
%e A331871 16: (oooo)
%e A331871 28: (oo(oo))
%e A331871 32: (ooooo)
%e A331871 38: (o(ooo))
%e A331871 49: ((oo)(oo))
%e A331871 56: (ooo(oo))
%e A331871 64: (oooooo)
%e A331871 76: (oo(ooo))
%e A331871 86: (o(o(oo)))
%e A331871 98: (o(oo)(oo))
%e A331871 106: (o(oooo))
%e A331871 112: (oooo(oo))
%e A331871 128: (ooooooo)
%e A331871 152: (ooo(ooo))
%e A331871 172: (oo(o(oo)))
%e A331871 196: (oo(oo)(oo))
%e A331871 The sequence of terms together with their prime indices begins:
%e A331871 1: {} 212: {1,1,16}
%e A331871 4: {1,1} 214: {1,28}
%e A331871 8: {1,1,1} 224: {1,1,1,1,1,4}
%e A331871 14: {1,4} 256: {1,1,1,1,1,1,1,1}
%e A331871 16: {1,1,1,1} 262: {1,32}
%e A331871 28: {1,1,4} 304: {1,1,1,1,8}
%e A331871 32: {1,1,1,1,1} 326: {1,38}
%e A331871 38: {1,8} 343: {4,4,4}
%e A331871 49: {4,4} 344: {1,1,1,14}
%e A331871 56: {1,1,1,4} 361: {8,8}
%e A331871 64: {1,1,1,1,1,1} 392: {1,1,1,4,4}
%e A331871 76: {1,1,8} 424: {1,1,1,16}
%e A331871 86: {1,14} 428: {1,1,28}
%e A331871 98: {1,4,4} 448: {1,1,1,1,1,1,4}
%e A331871 106: {1,16} 454: {1,49}
%e A331871 112: {1,1,1,1,4} 512: {1,1,1,1,1,1,1,1,1}
%e A331871 128: {1,1,1,1,1,1,1} 524: {1,1,32}
%e A331871 152: {1,1,1,8} 526: {1,56}
%e A331871 172: {1,1,14} 608: {1,1,1,1,1,8}
%e A331871 196: {1,1,4,4} 622: {1,64}
%t A331871 msQ[n_]:=n==1||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
%t A331871 Select[Range[1000],msQ]
%Y A331871 Not requiring local disjointness gives A291636.
%Y A331871 Not requiring lone-child avoidance gives A316495.
%Y A331871 A superset of A320269.
%Y A331871 These trees are counted by A331680.
%Y A331871 The semi-identity tree version is A331683.
%Y A331871 The version containing 2 is A331873.
%Y A331871 Cf. A001678, A007097, A050381, A061775, A196050, A302569, A302696, A316473, A316694, A316696, A316697, A331682, A331686, A331687, A331872, A331935.
%K A331871 nonn
%O A331871 1,2
%A A331871 _Gus Wiseman_, Feb 02 2020