A005517 -id:A005517 - cp's OEIS frontend

A347620 Position of Matula-Goebel number n among Matula-Goebel numbers sorted by number of vertices then numerically as in A061773.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 15, 16, 19, 17, 20, 21, 22, 23, 24, 38, 25, 39, 26, 27, 40, 28, 29, 41, 30, 42, 43, 31, 32, 44, 45, 33, 46, 34, 47, 86, 48, 49, 50, 51, 87, 52, 53, 35, 88, 89, 54, 55, 56, 36, 90, 57, 58, 91, 59, 92, 93, 37, 60

Offset: 1

Kevin Ryde, Sep 09 2021

nonn

This sequence is a permutation of the natural numbers, the inverse of A061773.
n = A005517(k) is the Matula-Goebel number of the first tree of k vertices so its position is immediately after all trees of 1..k-1 vertices so a(A005517(k)) = A087803(k-1) + 1.
n = A005518(k) is the last tree of k vertices so its position is a(A005518(k)) = A087803(k).

			Tree n=25 is the first of 7 vertices (A005517(7)=25), so its position is after the A087803(6)=37 trees of 1..6 vertices so a(25) = 38.
Tree n=27 is the next of 7 vertices (has A061775(27)=7) so it is next after position 38: a(27) = 39.

Cf. A061775 (number of vertices), A005517 (smallest), A005518 (largest), A087803 (number of trees).
Cf. A061773 (inverse).
Cf. A347540.

PARI
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a(n) = A087803(k-1) + s where s is the number of terms of A061775(1..n) equal to k, where k = A061775(n) is the number of vertices of n.

A235112 a(n) = the largest of the M-indices of the trees with n vertices.

Original entry on oeis.org

1, 2, 3, 7, 16, 32, 64, 152, 361, 1273, 4489, 22177, 109561, 735151

Offset: 1

Emeric Deutsch, Jan 03 2014

We define the M-index of a tree T to be the smallest of the Matula numbers of the rooted trees isomorphic (as a tree) to T. Example. The path tree P[5] = ABCDE has M-index 9. Indeed, there are 3 rooted trees isomorphic to P[5]: rooted at A, B, and C, respectively. Their Matula numbers are 11, 10, and 9, respectively. Consequently, the M-index of P[5] is 9.
a(n) = largest (= last) entry in row n of A235111.
It is conjectured that for n>=7 one has a(n) = A235120(n-6).
These numbers can be useful, for example, in the following situation. We intend to identify all trees that have 10 vertices and satisfy a certain property. Instead of scanning all rooted trees with Matula numbers from A005517(10)=125 to A005518(10)=219613, we do the scanning only for Matula numbers between 125 and a(10)=1273.

			a(4)=7. Indeed, there are 2 trees with 4 vertices: the path P[4] and the star S[3] with 3 edges. There are two rooted trees isomorphic to P[4]; they have Matula numbers 5 and 6; so the M-index is 5. There are two rooted trees isomorphic to S[3]; they have Matula numbers 7 and 8; so the M-index is 7. Max(5,7) = 7.

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011.
E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., 40, 2000, 113-116.
I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, On the multiplicative Wiener index and its possible chemical applications, Monatshefte f. Chemie, 131, 2000, 421-427.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A235111, A005518.

a(n) = A235111(n,A000055(n)).

a(13) from Emeric Deutsch, Feb 16 2014

a(14) from Emeric Deutsch, Mar 12 2014

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

Original entry on oeis.org

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

Offset: 1

Kevin Ryde, Mar 16 2022

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.

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

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

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

Column 1 of A245824.

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

PARI
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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

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

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

a(15) from Andrew Howroyd, Sep 17 2023

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A347620 Position of Matula-Goebel number n among Matula-Goebel numbers sorted by number of vertices then numerically as in A061773.

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A235112 a(n) = the largest of the M-indices of the trees with n vertices.

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

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