A331936 - OEIS

A331936

Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 144, 148, 152, 162, 169, 172, 178, 184, 192, 196, 202, 206, 208, 212, 214, 216, 224, 243, 244, 256, 262, 288

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OFFSET

1,2

COMMENTS

First differs from A331873 in lacking 69, the Matula-Goebel number of the tree ((o)((o)(o))).

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf.

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.

Consists of 1, 2, and all numbers equal to a power of 2 (other than 1) times a power of prime(j) for some j > 1 already in the sequence.

LINKS

Table of n, a(n) for n=1..60.

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

FORMULA

Intersection of A320230 and A331935.

EXAMPLE

The sequence of rooted trees ranked by this sequence together with their Matula-Goebel numbers begins:

1: o

2: (o)

4: (oo)

6: (o(o))

8: (ooo)

9: ((o)(o))

12: (oo(o))

14: (o(oo))

16: (oooo)

18: (o(o)(o))

24: (ooo(o))

26: (o(o(o)))

27: ((o)(o)(o))

28: (oo(oo))

32: (ooooo)

36: (oo(o)(o))

38: (o(ooo))

46: (o((o)(o)))

48: (oooo(o))

49: ((oo)(oo))

The sequence of terms together with their prime indices begins:

1: {} 52: {1,1,6} 152: {1,1,1,8}

2: {1} 54: {1,2,2,2} 162: {1,2,2,2,2}

4: {1,1} 56: {1,1,1,4} 169: {6,6}

6: {1,2} 64: {1,1,1,1,1,1} 172: {1,1,14}

8: {1,1,1} 72: {1,1,1,2,2} 178: {1,24}

9: {2,2} 74: {1,12} 184: {1,1,1,9}

12: {1,1,2} 76: {1,1,8} 192: {1,1,1,1,1,1,2}

14: {1,4} 81: {2,2,2,2} 196: {1,1,4,4}

16: {1,1,1,1} 86: {1,14} 202: {1,26}

18: {1,2,2} 92: {1,1,9} 206: {1,27}

24: {1,1,1,2} 96: {1,1,1,1,1,2} 208: {1,1,1,1,6}

26: {1,6} 98: {1,4,4} 212: {1,1,16}

27: {2,2,2} 104: {1,1,1,6} 214: {1,28}

28: {1,1,4} 106: {1,16} 216: {1,1,1,2,2,2}

32: {1,1,1,1,1} 108: {1,1,2,2,2} 224: {1,1,1,1,1,4}

36: {1,1,2,2} 112: {1,1,1,1,4} 243: {2,2,2,2,2}

38: {1,8} 122: {1,18} 244: {1,1,18}

46: {1,9} 128: {1,1,1,1,1,1,1} 256: {1,1,1,1,1,1,1,1}

48: {1,1,1,1,2} 144: {1,1,1,1,2,2} 262: {1,32}

49: {4,4} 148: {1,1,12} 288: {1,1,1,1,1,2,2}

MATHEMATICA

msQ[n_]:=n<=2||!PrimeQ[n]&&Length[DeleteCases[FactorInteger[n], {2, _}]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];

Select[Range[100], msQ]

CROSSREFS

A superset of A000079.

The non-lone-child-avoiding version is A320230.

The non-semi version is A320269.

These trees are counted by A331933.

Not requiring semi-achirality gives A331935.

The fully-achiral case is A331992.

Achiral trees are counted by A003238.

Numbers with at most one distinct odd prime factor are A070776.

Matula-Goebel numbers of achiral rooted trees are A214577.

Matula-Goebel numbers of semi-identity trees are A306202.

Numbers S with at most one distinct prime index in S are A331912.

Cf. A001678, A007097, A050381, A061775, A196050, A291636, A331784, A331873, A331914, A331934, A331965, A331967, A331991.

Sequence in context: A161819 A331935 A331873 * A364123 A244799 A347262

Adjacent sequences: A331933 A331934 A331935 * A331937 A331938 A331939

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 03 2020

STATUS

approved