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A331489 Matula-Goebel numbers of topologically series-reduced rooted trees. 6
1, 2, 7, 8, 16, 19, 28, 32, 43, 53, 56, 64, 76, 98, 107, 112, 128, 131, 152, 163, 172, 196, 212, 224, 227, 256, 263, 266, 304, 311, 343, 344, 383, 392, 424, 428, 443, 448, 512, 521, 524, 532, 577, 602, 608, 613, 652, 686, 688, 719, 722, 742, 751, 784, 848, 856 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

We say that a rooted tree is topologically series-reduced if no vertex (including the root) has degree 2.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.

LINKS

Table of n, a(n) for n=1..56.

Eric Weisstein's World of Mathematics, Series-reduced tree.

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

EXAMPLE

The sequence of all topologically series-reduced rooted trees together with their Matula-Goebel numbers begins:

    1: o

    2: (o)

    7: ((oo))

    8: (ooo)

   16: (oooo)

   19: ((ooo))

   28: (oo(oo))

   32: (ooooo)

   43: ((o(oo)))

   53: ((oooo))

   56: (ooo(oo))

   64: (oooooo)

   76: (oo(ooo))

   98: (o(oo)(oo))

  107: ((oo(oo)))

  112: (oooo(oo))

  128: (ooooooo)

  131: ((ooooo))

  152: (ooo(ooo))

  163: ((o(ooo)))

MATHEMATICA

nn=1000;

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

srQ[n_]:=Or[n==1, With[{m=primeMS[n]}, And[Length[m]>1, And@@srQ/@m]]];

Select[Range[nn], PrimeOmega[#]!=2&&And@@srQ/@primeMS[#]&]

CROSSREFS

Unlabeled rooted trees are counted by A000081.

Topologically series-reduced trees are counted by A000014.

Topologically series-reduced rooted trees are counted by A001679.

Labeled topologically series-reduced trees are counted by A005512.

Labeled topologically series-reduced rooted trees are counted by A060313.

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

Cf. A000669, A001678, A007097, A007821, A060356, A061775, A109082, A109129, A196050, A254382, A276625, A330943, A331490.

Sequence in context: A162664 A341706 A032689 * A300476 A213037 A287343

Adjacent sequences:  A331486 A331487 A331488 * A331490 A331491 A331492

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 20 2020

STATUS

approved

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Last modified June 16 14:35 EDT 2021. Contains 345057 sequences. (Running on oeis4.)