A206499 -id:A206499 - cp's OEIS frontend

A212618 Sum of the distances between all unordered pairs of vertices of degree 2 in the rooted tree with Matula-Goebel number n.

0, 0, 0, 0, 1, 1, 0, 0, 4, 4, 4, 0, 0, 0, 10, 0, 0, 2, 0, 1, 1, 10, 2, 0, 20, 2, 6, 0, 1, 6, 10, 0, 20, 1, 4, 2, 0, 0, 6, 1, 2, 0, 0, 4, 13, 6, 6, 0, 0, 14, 4, 0, 0, 6, 35, 0, 1, 6, 1, 6, 2, 20, 2, 0, 13, 13, 0, 0, 13, 1, 1, 2, 0, 2, 24, 0, 10, 3, 4, 1, 12, 6, 6, 0, 10, 0, 14, 4, 0, 13, 2, 2, 35, 13

Offset: 1

Emeric Deutsch, May 22 2012

nonn

The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m >= 2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

			a(11)=4 because the rooted tree with Matula-Goebel number 11 is the path tree A - B - C - D - E; the vertices of degree 2 are B, C, and D; we have dist(B,C)+dist(B,D)+dist(C,D) = 1+2+1 = 4.
a(987654321) = 68, as given by the Maple program; the reader can verify this on the rooted tree of Fig. 2 of the Deutsch reference.

Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
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.
A. Ilic and M. Ilic, Generalizations of Wiener polarity index and terminal Wiener index, arXiv:11106.2986 [math.CO], 2011-2012.
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. A206499, A212619, A212620, A212621, A212622, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

k := 2: with(numtheory): g := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then sort(expand(x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) = k then sort(expand(-x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) <> k-1 and bigomega(pi(n)) <> k then sort(expand(x*g(pi(n)))) elif bigomega(s(n)) = k-1 then sort(expand(1+g(r(n))+g(s(n)))) elif bigomega(s(n)) = k then sort(expand(-1+g(r(n))+g(s(n)))) else sort(g(r(n))+g(s(n))) end if end proc; with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then a(pi(n))+subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) = k then a(pi(n))-subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) <> k and bigomega(pi(n)) <> k-1 then a(pi(n)) elif bigomega(s(n)) = k-1 then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))+subs(x = 1, diff(g(r(n)), x))+subs(x = 1, diff(g(s(n)), x)) elif bigomega(s(n)) = k then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))-subs(x = 1, diff(g(r(n)), x))-subs(x = 1, diff(g(s(n)), x)) else a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x)) end if end proc: seq(a(n), n = 1 .. 120);

k = 2;
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
g[n_] := Which[n == 1, 0,
  PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1,
  Expand[x + x*g[PrimePi[n]]] ,
  PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k,
  Expand[-x + x*g[PrimePi[n]]],
  PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k - 1 &&
  PrimeOmega[PrimePi[n]] != k, Expand[x*g[PrimePi[n]]],
  PrimeOmega[s[n]] == k - 1, Expand[1 + g[r[n]] + g[s[n]]],
  PrimeOmega[s[n]] == k, Expand[-1 + g[r[n]] + g[s[n]]], True,
  g[r[n]] + g[s[n]]];
a[n_] := Which[n == 1, 0,
   PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1,
   a[PrimePi[n]] + (D[g[PrimePi[n]], x] /. x -> 1),
   PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k,
   a[PrimePi[n]] - (D[g[PrimePi[n]], x] /. x -> 1),
   PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k &&
    PrimeOmega[PrimePi[n]] != k - 1, a[PrimePi[n]],
   PrimeOmega[s[n]] == k - 1,
   a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) +
    (D[g[r[n]], x] /. x -> 1) + (D[g[s[n]], x] /. x -> 1) ,
    PrimeOmega[s[n]] == k,
   a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) -
    (D[g[r[n]], x] /. x -> 1) - (D[g[s[n]], x] /. x -> 1), True,
   a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) ];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 19 2024, after Maple code *)

We give recurrence formulas for the more general case of vertices of degree k (k>=2). Let bigomega(n) denote the number of prime divisors of n, counted with multiplicities. Let g(n)=g(n,k,x) be the generating polynomial of the vertices of degree k of the rooted tree with Matula-Goebel number n with respect to level. We have a(1) = 0; if n = prime(t) and bigomega(t)=k-1 then a(n) = a(t)+[dg(t)/dx]{x=1}; if n = prime(t) and bigomega(t)=k, then a(n) = a(t)-[dg(t)/dx]{x=1}; if n = prime(t) and bigomega(t) =/ k and =/ k-1, then a(n)= a(t); if n = r*s with r prime, s>=2, bigomega(s)=k-1, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]{x=1} +[dg(r)/dx]{x=1} +[dg(s)/dx]{x=1}; if n = r*s with r prime, s>=2, bigomega(s) =k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]{x=1} - [dg(r)/dx]{x=1} - [dg(s)/dx]{x=1}; if n = r*s with r prime, s>=2, bigomega(s) != k-1 and != k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1}.

A212619 Sum of the distances between all unordered pairs of vertices of degree 3 in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 4, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 3, 0

Offset: 1

Emeric Deutsch, May 22 2012

nonn

The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

			a(28)=1 because the rooted tree with Matula-Goebel number 28 is obtained by joining the trees I, I, and Y at  their roots; it has 2 vertices of degree 3 (the root and the center of Y), the distance between them is 1.
a(987654321) = 22, as given by the Maple program; the reader can verify this on the rooted tree of Fig. 2 of the Deutsch reference.

Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
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.
A. Ilic and M. Ilic, Generalizations of Wiener polarity index and terminal Wiener index, arXiv:11106.2986, 2011-2012.
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. A206499, A212618, A212620, A212621, A212622, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

k := 3: with(numtheory): g := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then sort(expand(x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) = k then sort(expand(-x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) <> k-1 and bigomega(pi(n)) <> k then sort(expand(x*g(pi(n)))) elif bigomega(s(n)) = k-1 then sort(expand(1+g(r(n))+g(s(n)))) elif bigomega(s(n)) = k then sort(expand(-1+g(r(n))+g(s(n)))) else sort(g(r(n))+g(s(n))) end if end proc; with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then a(pi(n))+subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) = k then a(pi(n))-subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) <> k and bigomega(pi(n)) <> k-1 then a(pi(n)) elif bigomega(s(n)) = k-1 then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))+subs(x = 1, diff(g(r(n)), x))+subs(x = 1, diff(g(s(n)), x)) elif bigomega(s(n)) = k then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))-subs(x = 1, diff(g(r(n)), x))-subs(x = 1, diff(g(s(n)), x)) else a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x)) end if end proc: seq(a(n), n = 1 .. 120);

k = 3;
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
g[n_] := Which[n == 1, 0, PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1, Expand[x + x*g[PrimePi[n]]], PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k, Expand[-x + x*g[PrimePi[n]]], PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k - 1 && PrimeOmega[PrimePi[n]] != k,  Expand[x*g[PrimePi[n]]], PrimeOmega[s[n]] == k - 1, Expand[1 + g[r[n]] + g[s[n]]], PrimeOmega[s[n]] == k, Expand[-1 + g[r[n]] + g[s[n]]], True, g[r[n]] + g[s[n]]];
a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1, a[PrimePi[n]] + (D[g[PrimePi[n]], x] /. x -> 1), PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k, a[PrimePi[n]] - (D[g[PrimePi[n]], x] /. x -> 1), PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k && PrimeOmega[PrimePi[n]] != k - 1, a[PrimePi[n]], PrimeOmega[s[n]] == k - 1, a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) + (D[g[r[n]], x] /. x -> 1) + (D[g[s[n]], x] /. x -> 1), PrimeOmega[s[n]] == k, a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) - (D[g[r[n]], x] /. x -> 1) - (D[g[s[n]], x] /. x -> 1), True, a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1)];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 19 2024, after Maple code *)

We give recurrence formulas for the more general case of vertices of degree k (k>=2). Let bigomega(n) denote the number of prime divisors of n, counted with multiplicities. Let g(n)=g(n,k,x) be the generating polynomial of the vertices of degree k of the rooted tree with Matula-Goebel number n with respect to level. We have a(1)=0; if n = prime(t) and bigomega(t) = k-1 then a(n) = a(t) +[dg(t)/dx]{x=1}; if n = prime(t) and bigomega(t) =k, then a(n) = a(t) - [dg(t)/dx]{x=1}; if n = prime(t) and bigomega(t) != k and != k-1, then a(n) = a(t); if n = r*s with r prime, s>=2, bigomega(s) =k-1, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]{x=1} +[dg(r)/dx]{x=1} +[dg(s)/dx]{x=1}; if n = r*s with r prime, s>=2, bigomega(s) =k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]{x=1} - [dg(r)/dx]{x=1} - [dg(s)/dx]{x=1}; if n = r*s with r prime, s>=2, bigomega(s) =/ k-1 and =/ k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1}.

A212628 Number of maximal independent vertex subsets in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 2, 2, 4, 4, 4, 3, 3, 3, 5, 2, 3, 5, 2, 4, 4, 5, 5, 3, 7, 5, 8, 3, 4, 6, 5, 2, 7, 4, 5, 5, 3, 3, 6, 4, 5, 5, 3, 5, 9, 8, 6, 3, 4, 8, 5, 5, 2, 9, 9, 3, 4, 6, 4, 6, 5, 7, 8, 2, 8, 8, 3, 4, 9, 6, 4, 5, 5, 5, 11, 3, 7, 8, 5, 4, 16, 6, 8, 5, 7, 5, 8, 5, 3, 10, 6, 8, 9, 9, 5, 3, 8, 5, 13, 8, 5, 6, 9, 5, 9, 3, 3, 9, 6, 10, 6, 3, 6, 5, 13, 6, 12, 5, 5, 6

Offset: 1

Emeric Deutsch, Jun 08 2012

nonn

A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent. An independent vertex subset S of a tree is said to be maximal if  every vertex that is not in S is joined by an edge to at least one vertex of S.
The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
Let A(n)=A(n,x), B(n)=B(n,x), C(n)=C(n,x) be the generating polynomial with respect to size of the maximal independent sets that contain the root, the maximal independent sets that do not contain the root, and the independent sets which are not maximal but become maximal if the root is removed, respectively. We have A(1)=x, B(1)=0, C(1)=1, A(t-th prime) = x[B(t) + C(t)], B(t-th prime) = A(t), C(t-th prime)=B(t), A(rs)=A(r)A(s)/x, B(rs)=B(r)B(s)+B(r)C(s)+B(s)C(r), C(rs)=C(r)C(s) (r,s>=2). The generating polynomial of the maximal independent vertex subsets with respect to size is P(n, x)=A(n,x)+B(n,x). Then a(n) = P(1,n). The Maple program is based on these relations.

			a(11)=4 because the rooted tree with Matula-Goebel number 11 is the path tree on 5 vertices R - A - B - C - D; the maximal independent vertex subsets are {R,C}, {A,C}, {A,D}, and {R,B,D}.

É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
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
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
H. S. Wilf, The number of maximal independent sets in a tree, SIAM J. Alg. Disc. Math., 7, 1986, 125-130.
Index entries for sequences related to Matula-Goebel numbers

Cf. A212627.

with(numtheory): P := proc (n) local r, s, A, B, C: r := n -> op(1, factorset(n)): s := n -> n/r(n): A := proc (n) if n = 1 then x elif bigomega(n) = 1 then x*(B(pi(n))+C(pi(n))) else A(r(n))*A(s(n))/x end if end proc: B := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then A(pi(n)) else sort(expand(B(r(n))*B(s(n))+B(r(n))*C(s(n))+B(s(n))*C(r(n)))) end if end proc: C := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then B(pi(n)) else expand(C(r(n))*C(s(n))) end if end proc: if n = 1 then x else sort(expand(A(n)+B(n))) end if end proc: seq(subs(x = 1, P(n)), n = 1 .. 120);
# For a more efficient calculation, the procedure P() could easily be simplified and optimized to yield A212628(n): remove "sort(expand...)" and replace x with 1 in appropriate places. - M. F. Hasler, Jan 06 2013

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
A[n_] := Which[n == 1, x, PrimeOmega[n] == 1, x*(B[PrimePi[n]] + c[PrimePi[n]]), True, A[r[n]]*A[s[n]]/x];
B[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, A[PrimePi[n]], True, B[r[n]]*B[s[n]] + B[r[n]]*c[s[n]] + B[s[n]]*c[r[n]]];
c[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, B[PrimePi[n]], True, c[r[n]]*c[s[n]]];
P[n_] := A[n] + B[n];
a[n_] := CoefficientList[P[n], x] // Total;
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 20 2024, after Maple code *)

a(n) = Sum_{k>=1} A212627(n,k).

A212629 Number of vertices in all maximal independent vertex subsets in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 3, 3, 6, 6, 4, 4, 9, 9, 9, 8, 8, 8, 14, 5, 8, 15, 5, 11, 11, 14, 15, 10, 22, 15, 25, 10, 11, 20, 14, 6, 22, 11, 17, 19, 10, 10, 20, 13, 15, 18, 10, 17, 33, 25, 20, 12, 13, 28, 17, 18, 6, 36, 32, 12, 13, 20, 11, 24, 19, 22, 29, 7, 31, 31, 10, 13, 33, 24, 13, 23, 18, 19, 45, 12, 26, 32

Offset: 1

Emeric Deutsch, Jun 08 2012

nonn

A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent. An independent vertex subset S of a tree is said to be maximal if  every vertex that is not in S is joined by an edge to at least one vertex of S.
The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
Let A(n)=A(n,x), B(n)=B(n,x), C(n)=C(n,x) be the generating polynomial with respect to size of the maximal independent sets that contain the root, the maximal independent sets that do not contain the root, and the independent sets which are not maximal but become maximal if the root is removed, respectively. We have: A(1)=x, B(1)=0, C(1)=1, A(t-th prime) = x[B(t) + C(t)], B(t-th prime) = A(t), C(t-th prime)=B(t), A(rs)=A(r)A(s)/x, B(rs)=B(r)B(s)+B(r)C(s)+B(s)C(r), C(rs)=C(r)C(s) (r,s>=2). The generating polynomial of the maximal independent vertex subsets with respect to size is P(n, x)=A(n,x)+B(n,x). Then a(n)=subs(x=1, dP(n,x)/dx). The Maple program is based on these relations.

			a(11)=9 because the rooted tree with Matula-Goebel number 11 is the path tree on 5 vertices R - A - B - C - D; the maximal independent vertex subsets are {R,C}, {A,C}, {A,D}, and {R,B,D}.

É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
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.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
H. S. Wilf, The number of maximal independent sets in a tree, SIAM J. Alg. Disc. Math., 7, 1986, 125-130.
Index entries for sequences related to Matula-Goebel numbers

Cf. A206499, A212627, A212618-A212632.

with(numtheory): P := proc (n) local r, s, A, B, C: r := n-> op(1, factorset(n)): s := n-> n/r(n): A := proc (n) if n = 1 then x elif bigomega(n) = 1 then x*(B(pi(n))+C(pi(n))) else A(r(n))*A(s(n))/x end if end proc: B := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then A(pi(n)) else sort(expand(B(r(n))*B(s(n))+B(r(n))*C(s(n))+B(s(n))*C(r(n)))) end if end proc: C := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then B(pi(n)) else expand(C(r(n))*C(s(n))) end if end proc: if n = 1 then x else sort(expand(A(n)+B(n))) end if end proc: seq(subs(x = 1, diff(P(n), x)), n = 1 .. 120);

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
A[n_] := Which[n == 1, x, PrimeOmega[n] == 1, x*(B[PrimePi[n]] + c[PrimePi[n]]), True, A[r[n]]*A[s[n]]/x];
B[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, A[PrimePi[n]], True, B[r[n]]*B[s[n]] + B[r[n]]*c[s[n]] + B[s[n]]*c[r[n]]];
c[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, B[PrimePi[n]], True, c[r[n]]*c[s[n]]];
P[n_] := A[n] + B[n];
a[n_] := D[P[n], x] /. x -> 1;
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 20 2024, after Maple code *)

a(n) = Sum(k*A212627(n,k), k>=1).

Sequences A212618-A212632 edited by M. Marcus and M. F. Hasler, Jan 06 2013

cp's OEIS Frontend

A212618 Sum of the distances between all unordered pairs of vertices of degree 2 in the rooted tree with Matula-Goebel number n.

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A212619 Sum of the distances between all unordered pairs of vertices of degree 3 in the rooted tree with Matula-Goebel number n.

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A212628 Number of maximal independent vertex subsets in the rooted tree with Matula-Goebel number n.

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A212629 Number of vertices in all maximal independent vertex subsets in the rooted tree with Matula-Goebel number n.

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