A288469 -id:A288469 - cp's OEIS frontend

A348959 Childless terminal Wiener index of the rooted tree with Matula-Goebel number n.

0, 0, 0, 2, 0, 3, 2, 6, 4, 4, 0, 8, 3, 8, 5, 12, 2, 10, 6, 10, 10, 5, 4, 15, 6, 10, 12, 16, 4, 12, 0, 20, 6, 10, 12, 18, 8, 15, 12, 18, 3, 19, 8, 12, 14, 12, 5, 24, 20, 14, 12, 19, 12, 21, 7, 26, 18, 12, 2, 21, 10, 6, 22, 30, 14, 14, 6, 20, 14, 22, 10, 28, 10

Offset: 1

Kevin Ryde, Nov 05 2021

nonn

This is a variation on the terminal Wiener index defined by Gutman, Furtula, and Petrović. Here terminal vertices are taken as the childless vertices, so a(n) is the sum of the path lengths between pairs of childless vertices.

This sequence differs from the free tree form A196055 when n is prime, since n prime means the root is degree 1 so is a terminal vertex for A196055 but not here.

Kevin Ryde, Table of n, a(n) for n = 1..10000
F. Goebel, On a 1-1-Correspondence between Rooted Trees and Natural Numbers, Journal of Combinatorial Theory, series B, volume 29, 1980, pages 141-143.
Ivan Gutman, Boris Furtula and Miroslav Petrović, Terminal Wiener Index, Journal of Mathematical Chemistry, volume 46, 2009, pages 522-531.
D. W. Matula, A Natural Rooted Tree Enumeration By Prime Factorization, SIAM Review, volume 10, number 2, April 1968, page 273 (also at JSTOR).
Kevin Ryde, PARI/GP Code and Notes
Index entries for sequences related to Matula-Goebel numbers

Cf. A196055 (free tree), A196048 (external path length), A109129 (childless vertices), A288469 (unplant).

Cf. A027746 (prime factorization).

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a(n) = Sum_{j=1..k} (a(primepi(p[j])) + E(p[j])*(C(n)-C(p[j]))), where n = p[1]*...*p[k] is the prime factorization of n with multiplicity (A027746), E(n) = A196048(n) is external path length, and C(n) = A109129(n) is number of childless vertices.
a(n) = A196055(n) - (A196048(n) if n prime).
a(n) = A196055(A288469(n)).

A328661 If n is the k-th composite number then a(n) = a(k), otherwise a(n) = n.

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 3, 1, 5, 11, 2, 13, 7, 3, 1, 17, 5, 19, 11, 2, 13, 23, 7, 3, 1, 17, 5, 29, 19, 31, 11, 2, 13, 23, 7, 37, 3, 1, 17, 41, 5, 43, 29, 19, 31, 47, 11, 2, 13, 23, 7, 53, 37, 3, 1, 17, 41, 59, 5, 61, 43, 29, 19, 31, 47, 67, 11, 2, 13, 71, 23, 73

Offset: 1

Rémy Sigrist, Oct 24 2019

nonn

			a(42) = a(28) = a(18) = a(10) = a(5) = 5.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

See A288469 and A328018 for similar sequences.

Cf. A002808, A006508.

k=0; for (n=1, #a=vector(73), print1 (a[n] = if (bigomega(n)>1, a[k++], n) ", "))

a(n) = 1 iff n belongs to A006508.

cp's OEIS Frontend

A348959 Childless terminal Wiener index of the rooted tree with Matula-Goebel number n.

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