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

1,2

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.

References

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

Links

Table of n, a(n) for n=1..15.

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

Formula

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

Example

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

Prog

(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

Crossrefs

Column 1 of A245824.

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

Sequence in context: A215493 A079309 A026630 * A034459 A120747 A229314

Adjacent sequences: A352453 A352454 A352455 * A352457 A352458 A352459

Keyword

nonn,more

Author

Kevin Ryde, Mar 16 2022

Extensions

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

a(15) from Andrew Howroyd, Sep 17 2023

Status

approved