A358552 Node-height of the rooted tree with Matula-Goebel number n. Number of nodes in the longest path from root to leaf.

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

Offset

1,2

Comments

Edge-height is given by A109082 (see formula).

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

Antti Karttunen, Table of n, a(n) for n = 1..100000

Formula

a(n) = A109082(n) + 1.

a(n) = A061775(n) - A358729(n). - Antti Karttunen, Oct 23 2023

Example

The Matula-Goebel number of ((ooo(o))) is 89, and it has node-height 4, so a(89) = 4.

Mathematica

MGTree[n_]:=If[n==1, {}, MGTree/@If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Table[Depth[MGTree[n]]-1, {n, 100}]

Prog

(PARI) A358552(n) = { my(v=factor(n)[, 1], d=0); while(#v, d++; v=fold(setunion, apply(p->factor(primepi(p))[, 1]~, v))); (1+d); }; \\ (after Kevin Ryde in A109082) - Antti Karttunen, Oct 23 2023
(Python)
from functools import lru_cache
from sympy import isprime, primepi, primefactors
@lru_cache(maxsize=None)
def A358552(n):
    if n == 1 : return 1
    if isprime(n): return 1+A358552(primepi(n))
    return max(A358552(p) for p in primefactors(n)) # Chai Wah Wu, Apr 15 2024

Crossrefs

Positions of first appearances are A007097.

This statistic is counted by A034781, ordered A080936.

The ordered version is A358379(n) + 1.

A000081 counts rooted trees, ordered A000108.

A055277 counts rooted trees by nodes and leaves, ordered A001263.

Other statistics: A061775 (nodes), A109082 (edge-height), A109129 (leaves), A196050 (edges), A342507 (internals).

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

Cf. A206487, A358576, A358589, A358592, A358729.

Sequence in context: A304088 A286597 A358667 * A317713 A341041 A361660

Adjacent sequences: A358549 A358550 A358551 * A358553 A358554 A358555

Keyword

nonn

Author

Gus Wiseman, Nov 26 2022

Extensions

Data section extended up to a(108) by Antti Karttunen, Oct 23 2023

Status

approved