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

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

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