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

%I A352456 #27 Dec 19 2024 11:46:19
%S A352456 1,4,14,49,301,1589,9761,51529,452411,3041573,23140153,143573641,
%T A352456 1260538619,8474639717,64474684537
%N A352456 Smallest Matula-Goebel number of a rooted binary tree (everywhere 0 or 2 children) of n childless vertices.
%C A352456 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.
%D A352456 Audace A. V. Dossou-Olory. The topological trees with extreme Matula numbers. J. Combin. Math. Combin. Comput., 115 (2020), 215-225.
%H A352456 Audace Amen Vioutou Dossou-Olory, <a href="https://arxiv.org/abs/1806.03995">The topological trees with extremal Matula numbers</a>, arXiv:1806.03995 [math.CO], 2018.
%H A352456 Kevin Ryde, <a href="/A352456/a352456.gp.txt">PARI/GP Code</a>
%H A352456 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A352456 a(n) = Min_{x+y=n} prime(a(x))*prime(a(y)).
%e A352456 For n = 6, the tree a(6) = 1589 is
%e A352456 .
%e A352456         *   root
%e A352456      /    \
%e A352456     *      *       6 childless
%e A352456    / \    / \      vertices "@"
%e A352456   @  @   *   *
%e A352456         / \ / \
%e A352456         @ @ @ @
%e A352456 .
%o A352456 (PARI) \\ See links.
%o A352456 (Python)
%o A352456 from sympy import prime
%o A352456 from itertools import count, islice
%o A352456 def agen(): # generator of terms
%o A352456     alst, plst = [0, 1], [0, 2]
%o A352456     yield 1
%o A352456     for n in count(2):
%o A352456         an = min(plst[x]*plst[n-x] for x in range(1, n//2+1))
%o A352456         yield an
%o A352456         alst.append(an)
%o A352456         plst.append(prime(an))
%o A352456 print(list(islice(agen(), 10))) # _Michael S. Branicky_, Mar 17 2022
%Y A352456 Column 1 of A245824.
%Y A352456 Cf. A111299 (all binary trees), A005517 (smallest all trees), A000040 (primes).
%K A352456 nonn,more
%O A352456 1,2
%A A352456 _Kevin Ryde_, Mar 16 2022
%E A352456 a(14) from _Michael S. Branicky_, Mar 17 2022
%E A352456 a(15) from _Andrew Howroyd_, Sep 17 2023