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

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

Author

Kevin Ryde, Jul 15 2022

Keywords

• nonn

Comments

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.

Examples

			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
		

Links

• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A000720, A001222 (bigomega), A354322 (distinct subtrees).

Cf. A291636 (indices of !=1).

Cf. A355661 (maximum children).

Programs

Maple

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

Mathematica

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 *)

PARI

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

Formula

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).