A331490 - OEIS

A331490

Matula-Goebel numbers of series-reduced rooted trees with more than two branches (of the root).

8, 16, 28, 32, 56, 64, 76, 98, 112, 128, 152, 172, 196, 212, 224, 256, 266, 304, 343, 344, 392, 424, 428, 448, 512, 524, 532, 602, 608, 652, 686, 688, 722, 742, 784, 848, 856, 896, 908, 931, 1024, 1048, 1052, 1064, 1204, 1216, 1244, 1304, 1372, 1376, 1444

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OFFSET

1,1

COMMENTS

We say that a rooted tree is (topologically) series-reduced if no vertex 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.

Also Matula-Goebel numbers of lone-child-avoiding rooted trees with more than two branches.

LINKS

Table of n, a(n) for n=1..51.

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

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 series-reduced rooted trees with more than two branches together with their Matula-Goebel numbers begins:

8: (ooo)

16: (oooo)

28: (oo(oo))

32: (ooooo)

56: (ooo(oo))

64: (oooooo)

76: (oo(ooo))

98: (o(oo)(oo))

112: (oooo(oo))

128: (ooooooo)

152: (ooo(ooo))

172: (oo(o(oo)))

196: (oo(oo)(oo))

212: (oo(oooo))

224: (ooooo(oo))

256: (oooooooo)

266: (o(oo)(ooo))

304: (oooo(ooo))

343: ((oo)(oo)(oo))

344: (ooo(o(oo)))

MATHEMATICA

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[1000], PrimeOmega[#]>2&&srQ[#]&]

CROSSREFS

These trees are counted by A331488.

Unlabeled rooted trees are counted by A000081.

Lone-child-avoiding rooted trees are counted by A001678.

Topologically series-reduced rooted trees are counted by A001679.

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

Matula-Goebel numbers of series-reduced rooted trees are A331489.

Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.

Sequence in context: A053093 A175086 A348231 * A039288 A045237 A255993

Adjacent sequences: A331487 A331488 A331489 * A331491 A331492 A331493

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 20 2020

STATUS

approved