A358578 - OEIS

A358578

Matula-Goebel numbers of rooted trees whose number of leaves equals their number of internal (non-leaf) nodes.

2, 6, 7, 18, 20, 21, 26, 34, 37, 43, 54, 60, 63, 67, 70, 78, 88, 91, 92, 95, 102, 111, 116, 119, 122, 129, 142, 146, 151, 162, 164, 173, 180, 181, 189, 200, 201, 202, 210, 227, 234, 236, 239, 245, 260, 264, 269, 273, 276, 278, 285, 306, 308, 314, 322, 333, 337

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Gus Wiseman, The first 64 rooted trees whose number of leaves equals their number of internal nodes.

FORMULA

A342507(a(n)) = A109129(a(n)).

EXAMPLE

The terms together with their corresponding rooted trees begin:

2: (o)

6: (o(o))

7: ((oo))

18: (o(o)(o))

20: (oo((o)))

21: ((o)(oo))

26: (o(o(o)))

34: (o((oo)))

37: ((oo(o)))

43: ((o(oo)))

54: (o(o)(o)(o))

60: (oo(o)((o)))

63: ((o)(o)(oo))

67: (((ooo)))

70: (o((o))(oo))

78: (o(o)(o(o)))

88: (ooo(((o))))

91: ((oo)(o(o)))

MATHEMATICA

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

Select[Range[337], Count[MGTree[#], {}, {0, Infinity}]==Count[MGTree[#], _[__], {0, Infinity}]&]

PROG

(Magma) PrimePi:=func< p |#PrimesUpTo(p)>; function C(n) if n eq 1 then return <1, 0>; end if; L:=0; I:=0; for t in Factorization(n) do for j in [1..t[2]]do c:=C(PrimePi(t[1]));

L+:=c[1]; I+:=c[2]; end for; end for; return <L, I+1>; end function; N:=150; seq:=[]; n:=1;

while #seq lt N do c:=C(n); if c[1] eq c[2] then Append(~seq, n); end if; n+:=1; end while;

seq; // Vincenzo Librandi, Mar 15 2026

CROSSREFS

These trees are counted by A185650, ordered A358579.

Height instead of leaves: A358576, counted by A358587, ordered A358588.

Height instead of internals: A358577, counted by A358589, ordered A358590.

Positions of 0's in A358580.

A000081 counts rooted trees, ordered A000108.

A034781 counts trees by nodes and height.

A055277 counts trees by nodes and leaves, ordered A001263.

MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.