A348066 Matula-Goebel tree number of tree n with all duplicate sibling subtrees reduced to one copy of each.

1, 2, 3, 2, 5, 6, 3, 2, 3, 10, 11, 6, 13, 6, 15, 2, 5, 6, 3, 10, 3, 22, 5, 6, 5, 26, 3, 6, 29, 30, 31, 2, 33, 10, 15, 6, 13, 6, 39, 10, 41, 6, 13, 22, 15, 10, 47, 6, 3, 10, 15, 26, 3, 6, 55, 6, 3, 58, 11, 30, 13, 62, 3, 2, 65, 66, 5, 10, 15, 30, 29, 6, 5, 26

Offset: 1

Author

Kevin Ryde, Oct 01 2021

Keywords

• nonn

Comments

Here a subtree means a vertex and all its descendants (children, grandchildren, etc.).

Any subtree which is a duplicate of a sibling subtree is deleted and this is repeated until no duplicates exist. The order of deletions does not affect the result.

Or equivalently, duplicates among sibling subtrees are considered after each subtree has itself been reduced.

The resulting tree a(n) has no duplicates, which means asymmetric so that a(n) is a term of A276625. Reduction is no-change a(n) = n if and only if n is in A276625.

This sequence is a divisibility sequence since the prime factors of a(m*n) are the union of the prime factors of a(m) and a(n), so that a(m*n) is divisible by a(n).

Examples

			In the following tree n=42, E and F are (singleton) subtrees of B and are the same so reduce to E alone. With that change, subtrees A,D and B,E are the same, so reduce to A,D alone.
  tree n=42    tree a(42) = 6
      R___         R         root R
     / \  \        | \
    A   B  C       A  C
    |   |\         |
    D   E F        D
		

Links

• Kevin Ryde, Table of n, a(n) for n = 1..10000
• Index to divisibility sequences
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A276625 (asymmetric trees), A007947.

Programs

PARI

a(n) = vecprod(Set([prime(self()(primepi(p))) | p<-factor(n)[,1]]));

Formula

a(n) = Product of distinct prime(a(primepi(p))), over p the prime factors of n.