A354322 -id:A354322 - 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).

A355662 Smallest number of children of any vertex which has children, in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

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

Offset: 1

Kevin Ryde, Jul 15 2022

nonn

Record highs are at a(2^k) = k which is a root with k singleton children.

If n is prime then the root has a single child so that a(n) = 1.

			For n=31972, the tree is as follows and vertex 1007 has 2 children which is the least among the vertices which have children, so a(31972) = 2.
    31972  root
   / |  \
  1  1  1007      Tree n=31972 and its
       /    \     subtree numbers.
      8      16
     /|\    // \\
    1 1 1  1 1 1 1

Index entries for sequences related to Matula-Goebel numbers

Cf. A000720, A001222 (bigomega), A354322 (distinct subtrees).
Cf. A291636 (indices of !=1).
Cf. A355661 (maximum children).

a:= proc(n) option remember; uses numtheory;
      min(bigomega(n), map(p-> a(pi(p)), factorset(n) minus {2})[])
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 15 2022

a[n_] := a[n] = Min[Join[{PrimeOmega[n]}, a /@ PrimePi @ Select[ FactorInteger[n][[All, 1]], #>2&]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 08 2022 *)

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

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

a(n) = Min_{s>=2 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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