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 A331965 #17 Jun 25 2021 23:21:05
%S A331965 1,4,8,14,16,28,32,38,56,64,76,86,106,112,128,133,152,172,212,214,224,
%T A331965 256,262,266,301,304,326,344,371,424,428,448,512,524,526,532,602,608,
%U A331965 622,652,688,742,749,766,817,848,856,886,896,917,1007,1024,1048,1052
%N A331965 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.
%C A331965 First differs from A331683 in having 133, the Matula-Goebel number of the tree ((oo)(ooo)).
%C A331965 Lone-child-avoiding means there are no unary branchings.
%C A331965 In a semi-identity tree, the non-leaf branches of any given vertex are all distinct.
%C A331965 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 A331965 Consists of one, and all composite numbers that are n times a power of two, where n is a squarefree number whose prime indices already belong to the sequence, and a prime index of n is a number m such that prime(m) divides n. [Clarified by _Peter Munn_ and _Gus Wiseman_, Jun 24 2021]
%H A331965 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>
%H A331965 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A331965 Intersection of A291636 and A306202.
%e A331965 The sequence of all lone-child-avoiding rooted semi-identity trees together with their Matula-Goebel numbers begins:
%e A331965 1: o
%e A331965 4: (oo)
%e A331965 8: (ooo)
%e A331965 14: (o(oo))
%e A331965 16: (oooo)
%e A331965 28: (oo(oo))
%e A331965 32: (ooooo)
%e A331965 38: (o(ooo))
%e A331965 56: (ooo(oo))
%e A331965 64: (oooooo)
%e A331965 76: (oo(ooo))
%e A331965 86: (o(o(oo)))
%e A331965 106: (o(oooo))
%e A331965 112: (oooo(oo))
%e A331965 128: (ooooooo)
%e A331965 133: ((oo)(ooo))
%e A331965 152: (ooo(ooo))
%e A331965 172: (oo(o(oo)))
%e A331965 212: (oo(oooo))
%e A331965 214: (o(oo(oo)))
%e A331965 The sequence of terms together with their prime indices begins:
%e A331965 1: {} 224: {1,1,1,1,1,4}
%e A331965 4: {1,1} 256: {1,1,1,1,1,1,1,1}
%e A331965 8: {1,1,1} 262: {1,32}
%e A331965 14: {1,4} 266: {1,4,8}
%e A331965 16: {1,1,1,1} 301: {4,14}
%e A331965 28: {1,1,4} 304: {1,1,1,1,8}
%e A331965 32: {1,1,1,1,1} 326: {1,38}
%e A331965 38: {1,8} 344: {1,1,1,14}
%e A331965 56: {1,1,1,4} 371: {4,16}
%e A331965 64: {1,1,1,1,1,1} 424: {1,1,1,16}
%e A331965 76: {1,1,8} 428: {1,1,28}
%e A331965 86: {1,14} 448: {1,1,1,1,1,1,4}
%e A331965 106: {1,16} 512: {1,1,1,1,1,1,1,1,1}
%e A331965 112: {1,1,1,1,4} 524: {1,1,32}
%e A331965 128: {1,1,1,1,1,1,1} 526: {1,56}
%e A331965 133: {4,8} 532: {1,1,4,8}
%e A331965 152: {1,1,1,8} 602: {1,4,14}
%e A331965 172: {1,1,14} 608: {1,1,1,1,1,8}
%e A331965 212: {1,1,16} 622: {1,64}
%e A331965 214: {1,28} 652: {1,1,38}
%t A331965 csiQ[n_]:=n==1||!PrimeQ[n]&&FreeQ[FactorInteger[n],{_?(#>2&),_?(#>1&)}]&&And@@csiQ/@PrimePi/@First/@FactorInteger[n];
%t A331965 Select[Range[100],csiQ]
%Y A331965 The non-semi case is {1}.
%Y A331965 Not requiring lone-child-avoidance gives A306202.
%Y A331965 The locally disjoint version is A331683.
%Y A331965 These trees are counted by A331966.
%Y A331965 The semi-lone-child-avoiding case is A331994.
%Y A331965 Matula-Goebel numbers of rooted identity trees are A276625.
%Y A331965 Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
%Y A331965 Semi-identity trees are counted by A306200.
%Y A331965 Cf. A001678, A004111, A007097, A061775, A122132, A196050, A300660, A316694, A320269, A331681, A331686, A331875, A331936, A331937, A331993.
%K A331965 nonn
%O A331965 1,2
%A A331965 _Gus Wiseman_, Feb 04 2020