A352456 Smallest Matula-Goebel number of a rooted binary tree (everywhere 0 or 2 children) of n childless vertices.

1, 4, 14, 49, 301, 1589, 9761, 51529, 452411, 3041573, 23140153, 143573641, 1260538619, 8474639717, 64474684537

Offset: 1

Author

Kevin Ryde, Mar 16 2022

Keywords

• nonn
• more

Comments

In the formula below, the two subtrees of the root have x and y childless vertices. The minimum Matula-Goebel number for that partition uses the minimum numbers for each subtree. The question is then which x+y partition is the overall minimum.

Examples

			For n = 6, the tree a(6) = 1589 is
.
        *   root
     /    \
    *      *       6 childless
   / \    / \      vertices "@"
  @  @   *   *
        / \ / \
        @ @ @ @
.
		

References

• Audace A. V. Dossou-Olory. The topological trees with extreme Matula numbers. J. Combin. Math. Combin. Comput., 115 (2020), 215-225.

Links

• Audace Amen Vioutou Dossou-Olory, The topological trees with extremal Matula numbers, arXiv:1806.03995 [math.CO], 2018.
• Kevin Ryde, PARI/GP Code
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Column 1 of A245824.

Cf. A111299 (all binary trees), A005517 (smallest all trees), A000040 (primes).

Programs

PARI

\\ See links.

Python

from sympy import prime
from itertools import count, islice
def agen(): # generator of terms
    alst, plst = [0, 1], [0, 2]
    yield 1
    for n in count(2):
        an = min(plst[x]*plst[n-x] for x in range(1, n//2+1))
        yield an
        alst.append(an)
        plst.append(prime(an))
print(list(islice(agen(), 10))) # Michael S. Branicky, Mar 17 2022

Formula

a(n) = Min_{x+y=n} prime(a(x))*prime(a(y)).

Extensions

a(14) from Michael S. Branicky, Mar 17 2022

a(15) from Andrew Howroyd, Sep 17 2023