OFFSET
1,8
COMMENTS
Node-height is the number of nodes in the longest path from root to leaf.
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.
Because the number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n, i.e., A317713(n) (= 1+A324923(n)), is always at least one larger than the depth of the same tree (= A109082(n)), it follows that a(n) >= A366386(n) for all n. - Antti Karttunen, Oct 23 2023
LINKS
FORMULA
EXAMPLE
The tree (oo(oo(o))) with Matula-Goebel number 148 has 8 nodes and node-height 4, so a(148) = 4.
MATHEMATICA
MGTree[n_]:=If[n==1, {}, MGTree/@Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Count[MGTree[n], _, {0, Infinity}]-(Depth[MGTree[n]]-1), {n, 100}]
PROG
(PARI)
A061775(n) = if(1==n, 1, if(isprime(n), 1+A061775(primepi(n)), {my(pfs, t, i); pfs=factor(n); pfs[, 1]=apply(t->A061775(t), pfs[, 1]); (1-bigomega(n)) + sum(i=1, omega(n), pfs[i, 1]*pfs[i, 2])}));
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 program given in A109082 by Kevin Ryde, Sep 21 2020)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 29 2022
EXTENSIONS
Data section extended up to a(108) by Antti Karttunen, Oct 23 2023
STATUS
approved