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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). 13
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 (list; graph; refs; listen; history; text; internal format)
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 * A244799 A316350 A071562

Adjacent sequences:  A331933 A331934 A331935 * A331937 A331938 A331939

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 03 2020

STATUS

approved

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Last modified April 5 05:06 EDT 2020. Contains 333238 sequences. (Running on oeis4.)