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

1,4

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.

Links

Table of n, a(n) for n=1..87.

Index entries for sequences related to Matula-Goebel numbers

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).

Example

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

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 *)

Prog

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

Crossrefs

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

Cf. A291636 (indices of !=1).

Cf. A355661 (maximum children).

Sequence in context: A095025 A274382 A318997 * A069897 A257242 A337908

Adjacent sequences: A355659 A355660 A355661 * A355663 A355664 A355665

Keyword

nonn

Author

Kevin Ryde, Jul 15 2022

Status

approved