A331871 - OEIS

%I #9 Feb 03 2020 22:17:51

%S 1,4,8,14,16,28,32,38,49,56,64,76,86,98,106,112,128,152,172,196,212,

%T 214,224,256,262,304,326,343,344,361,392,424,428,448,454,512,524,526,

%U 608,622,652,686,688,722,766,784,848,856,886,896,908,1024,1042,1048,1052

%N Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.

%C First differs from A320269 in having 1589, the Matula-Goebel number of the tree ((oo)((oo)(oo))).

%C First differs from A331683 in having 49.

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

%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 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 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 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 Intersection of A291636 and A316495.

%e The sequence of all lone-child-avoiding locally disjoint rooted 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    49: ((oo)(oo))

%e    56: (ooo(oo))

%e    64: (oooooo)

%e    76: (oo(ooo))

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

%e    98: (o(oo)(oo))

%e   106: (o(oooo))

%e   112: (oooo(oo))

%e   128: (ooooooo)

%e   152: (ooo(ooo))

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

%e   196: (oo(oo)(oo))

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

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

%e      4: {1,1}               214: {1,28}

%e      8: {1,1,1}             224: {1,1,1,1,1,4}

%e     14: {1,4}               256: {1,1,1,1,1,1,1,1}

%e     16: {1,1,1,1}           262: {1,32}

%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}               343: {4,4,4}

%e     49: {4,4}               344: {1,1,1,14}

%e     56: {1,1,1,4}           361: {8,8}

%e     64: {1,1,1,1,1,1}       392: {1,1,1,4,4}

%e     76: {1,1,8}             424: {1,1,1,16}

%e     86: {1,14}              428: {1,1,28}

%e     98: {1,4,4}             448: {1,1,1,1,1,1,4}

%e    106: {1,16}              454: {1,49}

%e    112: {1,1,1,1,4}         512: {1,1,1,1,1,1,1,1,1}

%e    128: {1,1,1,1,1,1,1}     524: {1,1,32}

%e    152: {1,1,1,8}           526: {1,56}

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

%e    196: {1,1,4,4}           622: {1,64}

%t msQ[n_]:=n==1||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];

%t Select[Range[1000],msQ]

%Y Not requiring local disjointness gives A291636.

%Y Not requiring lone-child avoidance gives A316495.

%Y A superset of A320269.

%Y These trees are counted by A331680.

%Y The semi-identity tree version is A331683.

%Y The version containing 2 is A331873.

%Y Cf. A001678, A007097, A050381, A061775, A196050, A302569, A302696, A316473, A316694, A316696, A316697, A331682, A331686, A331687, A331872, A331935.

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 02 2020