A355662 -id:A355662 - cp's OEIS frontend

A355661 Largest number of children of any vertex in the rooted tree with Matula-Goebel number n.

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

Offset: 1

Kevin Ryde, Jul 14 2022

nonn

Record highs are at a(2^k) = k which is a root with k singleton children.
A new root above a tree has a single child (the old root) so no change to the largest number of children, except when above a singleton, so that a(prime(n)) = a(n) for n >= 2.
Terms a(n) <= 1 are paths down (all vertices 0 or 1 children), which are the primeth recurrence n = A007097.

			For n=629, tree 629 is as follows and vertex 12 has 3 children which is the most of any vertex so that a(629) = 3.
      629  root
     /   \
    7     12     tree n=629 and its
    |     /|\    subtree numbers
    4    1 1 2
   / \       |
  1   1      1

Cf. A001222 (bigomega), A354322 (distinct subtrees).
Cf. A007097 (indices of <=1).
Cf. A355662 (minimum children).

a:= proc(n) option remember; uses numtheory;
      max(bigomega(n), map(p-> a(pi(p)), factorset(n))[])
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 14 2022

nn = 105; a[1] = 0; a[2] = 1; Do[a[n] = Max@ Append[Map[a[PrimePi[#]] &, FactorInteger[n][[All, 1]]], PrimeOmega[n]], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Jul 14 2022 *)

a(n) = my(f=factor(n)); vecmax(concat(vecsum(f[,2]), [self()(primepi(p)) |p<-f[,1]]));

a(n) = max(bigomega(n), {a(primepi(p)) | p prime factor of n}).

a(n) = Max_{s in row n of A354322} bigomega(s).

cp's OEIS Frontend

A355661 Largest number of children of any vertex in the rooted tree with Matula-Goebel number n.

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