A342507 Number of internal nodes in rooted tree with Matula-Goebel number n.

0, 1, 2, 1, 3, 2, 2, 1, 3, 3, 4, 2, 3, 2, 4, 1, 3, 3, 2, 3, 3, 4, 4, 2, 5, 3, 4, 2, 4, 4, 5, 1, 5, 3, 4, 3, 3, 2, 4, 3, 4, 3, 3, 4, 5, 4, 5, 2, 3, 5, 4, 3, 2, 4, 6, 2, 3, 4, 4, 4, 4, 5, 4, 1, 5, 5, 3, 3, 5, 4, 4, 3, 4, 3, 6, 2, 5, 4, 5, 3, 5, 4, 5, 3, 5, 3, 5, 4, 3, 5, 4, 4, 6, 5, 4, 2, 6, 3, 6, 5

Offset

1,3

Comments

The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product_{T_i} prime(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it. (Cf. A061773 for illustration.)

Links

François Marques, Table of n, a(n) for n = 1..10000

Index entries for sequences related to Matula-Goebel numbers

Formula

a(1)=0 and a(n) = A061775(n) - A109129(n) for n > 1.

Example

a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = 1 because the rooted tree with Matula-Goebel number 2^m is the star tree with m edges.

Mathematica

MGTree[n_]:=If[n==1, {}, MGTree/@Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Count[MGTree[n], _[__], {0, Infinity}], {n, 100}] (* Gus Wiseman, Nov 28 2022 *)

Prog

(PARI) A342507(n) = if( n==1, 0, my(f=factor(n)); 1+sum(k=1, matsize(f)[1], A342507(primepi(f[k, 1]))*f[k, 2]));

Crossrefs

Other statistics are: A061775 (nodes), A109082 (edge-height), A109129 (leaves), A196050 (edges), A358552 (node-height).

An ordered version is A358553.

Positions of first appearances are A358554.

A000081 counts rooted trees, ordered A000108.

A358575 counts rooted trees by nodes and internals.

Cf. A000040, A000720, A001222, A007097, A056239, A112798.

Cf. A034781, A055277, A206487, A358576, A358578, A358592.

Sequence in context: A306467 A157810 A072339 * A261337 A374998 A375000

Adjacent sequences: A342504 A342505 A342506 * A342508 A342509 A342510

Keyword

nonn

Author

François Marques, Mar 14 2021

Status

approved