A331994 - OEIS

A331994

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

%I #9 Feb 08 2020 11:46:36

%S 1,2,4,6,8,12,14,16,21,24,26,28,32,38,39,42,48,52,56,57,64,74,76,78,

%T 84,86,91,96,104,106,111,112,114,128,129,133,146,148,152,156,159,168,

%U 172,178,182,192,202,208,212,214,219,222,224,228,247,256,258,259,262

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

%C Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf.

%C In a semi-identity tree, the non-leaf branches of any given vertex are 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, two, and all numbers that can be written as a power of two (other than 2) times a squarefree number 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 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 A306202 and A331935.

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

%e 1: o

%e 2: (o)

%e 4: (oo)

%e 6: (o(o))

%e 8: (ooo)

%e 12: (oo(o))

%e 14: (o(oo))

%e 16: (oooo)

%e 21: ((o)(oo))

%e 24: (ooo(o))

%e 26: (o(o(o)))

%e 28: (oo(oo))

%e 32: (ooooo)

%e 38: (o(ooo))

%e 39: ((o)(o(o)))

%e 42: (o(o)(oo))

%e 48: (oooo(o))

%e 52: (oo(o(o)))

%e 56: (ooo(oo))

%e 57: ((o)(ooo))

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

%e 1: {} 64: {1,1,1,1,1,1} 159: {2,16}

%e 2: {1} 74: {1,12} 168: {1,1,1,2,4}

%e 4: {1,1} 76: {1,1,8} 172: {1,1,14}

%e 6: {1,2} 78: {1,2,6} 178: {1,24}

%e 8: {1,1,1} 84: {1,1,2,4} 182: {1,4,6}

%e 12: {1,1,2} 86: {1,14} 192: {1,1,1,1,1,1,2}

%e 14: {1,4} 91: {4,6} 202: {1,26}

%e 16: {1,1,1,1} 96: {1,1,1,1,1,2} 208: {1,1,1,1,6}

%e 21: {2,4} 104: {1,1,1,6} 212: {1,1,16}

%e 24: {1,1,1,2} 106: {1,16} 214: {1,28}

%e 26: {1,6} 111: {2,12} 219: {2,21}

%e 28: {1,1,4} 112: {1,1,1,1,4} 222: {1,2,12}

%e 32: {1,1,1,1,1} 114: {1,2,8} 224: {1,1,1,1,1,4}

%e 38: {1,8} 128: {1,1,1,1,1,1,1} 228: {1,1,2,8}

%e 39: {2,6} 129: {2,14} 247: {6,8}

%e 42: {1,2,4} 133: {4,8} 256: {1,1,1,1,1,1,1,1}

%e 48: {1,1,1,1,2} 146: {1,21} 258: {1,2,14}

%e 52: {1,1,6} 148: {1,1,12} 259: {4,12}

%e 56: {1,1,1,4} 152: {1,1,1,8} 262: {1,32}

%e 57: {2,8} 156: {1,1,2,6} 266: {1,4,8}

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

%t Select[Range[100],scsiQ]

%Y The locally disjoint version is A331681.

%Y The enumeration of these trees by vertices is A331993.

%Y Semi-identity trees are A306200.

%Y MG-numbers of rooted identity trees are A276625.

%Y MG-numbers of lone-child-avoiding rooted identity trees are {1}.

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

%Y MG-numbers of semi-identity trees are A306202.

%Y MG-numbers of semi-lone-child-avoiding rooted trees are A331935.

%Y MG-numbers of semi-lone-child-avoiding rooted identity trees are A331963.

%Y MG-numbers of lone-child-avoiding rooted semi-identity trees are A331965.

%Y Cf. A004111, A007097, A061775, A122132, A196050, A320269, A331683, A331873, A331934, A331936, A331964, A331966.

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 05 2020