%I #17 Jun 25 2021 23:21:05

%S 1,4,8,14,16,28,32,38,56,64,76,86,106,112,128,133,152,172,212,214,224,

%T 256,262,266,301,304,326,344,371,424,428,448,512,524,526,532,602,608,

%U 622,652,688,742,749,766,817,848,856,886,896,917,1007,1024,1048,1052

%N Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.

%C First differs from A331683 in having 133, the Matula-Goebel number of the tree ((oo)(ooo)).

%C Lone-child-avoiding means there are no unary branchings.

%C In a semi-identity tree, the non-leaf branches of any given vertex are all distinct.

%C 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 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 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 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F Intersection of A291636 and A306202.

%e The sequence of all lone-child-avoiding rooted semi-identity trees together with their Matula-Goebel numbers begins:

%e 1: o

%e 4: (oo)

%e 8: (ooo)

%e 14: (o(oo))

%e 16: (oooo)

%e 28: (oo(oo))

%e 32: (ooooo)

%e 38: (o(ooo))

%e 56: (ooo(oo))

%e 64: (oooooo)

%e 76: (oo(ooo))

%e 86: (o(o(oo)))

%e 106: (o(oooo))

%e 112: (oooo(oo))

%e 128: (ooooooo)

%e 133: ((oo)(ooo))

%e 152: (ooo(ooo))

%e 172: (oo(o(oo)))

%e 212: (oo(oooo))

%e 214: (o(oo(oo)))

%e The sequence of terms together with their prime indices begins:

%e 1: {} 224: {1,1,1,1,1,4}

%e 4: {1,1} 256: {1,1,1,1,1,1,1,1}

%e 8: {1,1,1} 262: {1,32}

%e 14: {1,4} 266: {1,4,8}

%e 16: {1,1,1,1} 301: {4,14}

%e 28: {1,1,4} 304: {1,1,1,1,8}

%e 32: {1,1,1,1,1} 326: {1,38}

%e 38: {1,8} 344: {1,1,1,14}

%e 56: {1,1,1,4} 371: {4,16}

%e 64: {1,1,1,1,1,1} 424: {1,1,1,16}

%e 76: {1,1,8} 428: {1,1,28}

%e 86: {1,14} 448: {1,1,1,1,1,1,4}

%e 106: {1,16} 512: {1,1,1,1,1,1,1,1,1}

%e 112: {1,1,1,1,4} 524: {1,1,32}

%e 128: {1,1,1,1,1,1,1} 526: {1,56}

%e 133: {4,8} 532: {1,1,4,8}

%e 152: {1,1,1,8} 602: {1,4,14}

%e 172: {1,1,14} 608: {1,1,1,1,1,8}

%e 212: {1,1,16} 622: {1,64}

%e 214: {1,28} 652: {1,1,38}

%t csiQ[n_]:=n==1||!PrimeQ[n]&&FreeQ[FactorInteger[n],{_?(#>2&),_?(#>1&)}]&&And@@csiQ/@PrimePi/@First/@FactorInteger[n];

%t Select[Range[100],csiQ]

%Y The non-semi case is {1}.

%Y Not requiring lone-child-avoidance gives A306202.

%Y The locally disjoint version is A331683.

%Y These trees are counted by A331966.

%Y The semi-lone-child-avoiding case is A331994.

%Y Matula-Goebel numbers of rooted identity trees are A276625.

%Y Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.

%Y Semi-identity trees are counted by A306200.

%Y Cf. A001678, A004111, A007097, A061775, A122132, A196050, A300660, A316694, A320269, A331681, A331686, A331875, A331936, A331937, A331993.

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 04 2020