A184160 -id:A184160 - cp's OEIS frontend

A206491 Irregular triangle read by rows: T(n,k) is the number of root subtrees with k nodes in the rooted tree having Matula-Goebel number n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 3, 3, 2, 1, 1, 4, 6, 4, 1, 1, 1, 1, 2, 1, 1, 3, 5, 5, 3, 1, 1, 1, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 2, 4, 4, 3, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 3, 2, 1, 1, 4, 7, 7, 4, 1

Offset: 1

Emeric Deutsch, May 08 2012

A root subtree of a rooted tree G is a subtree of G containing the root.
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.
Number of entries in row n = A061775(n).
Sum of entries in row n = A184160(n).
For the number of all subtrees of a given size, see A212620.

			Row 7 is 1,1,2,1 because the rooted tree with Matula-Goebel number 7 is Y; its five root subtrees have 1, 2, 3, 3, and 4 nodes.

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 Y-N. 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 Review, 10, 1968, 273.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Index entries for sequences related to Matula-Goebel numbers

Cf. A061775, A184160, A212620

with(numtheory): V := 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 1 elif bigomega(n) = 1 then 1+V(pi(n)) else V(r(n))+V(s(n))-1 end if end proc: R := proc (n, k) 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 and k = 1 then 1 elif n = 1 and 1 < k then 0 elif bigomega(n) = 1 and k = 1 then 1 elif bigomega(n) = 1 then R(pi(n), k-1) else add(R(r(n), j)*R(s(n), k+1-j), j = 1 .. k) end if end proc: for n to 40 do seq(R(n, k), k = 1 .. V(n)) end do; # yields sequence  in triangular form

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
V[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + V[PrimePi[n]], True, V[r[n]] + V[s[n]] - 1];
R[n_, k_] := Which[n == 1 && k == 1, 1, n == 1 && 1 < k, 0, PrimeOmega[n] == 1 && k == 1, 1, PrimeOmega[n] == 1, R[PrimePi[n], k-1], True, Sum[R[r[n], j]*R[s[n], k+1-j], {j, 1, k}]];
Table[R[n, k], {n, 1, 40}, {k, 1, V[n]}] // Flatten (* Jean-François Alcover, Oct 13 2024, after Emeric Deutsch *)

A184161 Number of subtrees in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 3, 6, 6, 10, 10, 11, 11, 15, 15, 15, 17, 17, 17, 21, 20, 17, 25, 20, 24, 24, 21, 25, 30, 28, 25, 36, 28, 24, 34, 21, 37, 28, 24, 32, 44, 30, 30, 34, 41, 25, 40, 28, 32, 48, 36, 34, 55, 37, 45, 32, 40, 37, 64, 36, 49, 41, 34, 24, 59, 44, 28, 57, 70, 44, 44, 30, 37, 48, 53, 41, 81, 40, 44, 63, 49, 41, 56, 32, 74

Offset: 1

Emeric Deutsch, Oct 19 2011

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(4) = 6 because the rooted tree with Matula-Goebel number 4 is V; it has 6 subtrees (three 1-vertex subtrees, two 1-edge subtrees, and the tree itself).

É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint 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.
Index entries for sequences related to Matula-Goebel numbers

Cf. A184160 (subtrees containing the root), A184164 (numbers not occurring as terms).

with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then 1+b(pi(n)) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n))+b(pi(n))+1 else a(r(n))+a(s(n))+(b(r(n))-1)*(b(s(n))-1)-1 end if end proc: seq(a(n), n = 1 .. 80);

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
b[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + b[PrimePi[n]], True, b[r[n]]*b[s[n]]];
a[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, a[PrimePi[n]] + b[PrimePi[n]] + 1, True, a[r[n]] + a[s[n]] + (b[r[n]] - 1)*(b[s[n]] - 1) - 1];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 24 2024, after Maple code *)

Let b(n)=A184160(n) denote the number of those subtrees of the rooted tree with Matula-Goebel number n that contain the root. Then a(1)=1; if n=prime(t) (=the t=th prime), then a(n)=1+a(t)+b(t); if n=r*s (r,s,>=2), then a(n)=a(r)+a(s)+(b(r)-1)*(b(s)-1)-1. The Maple program is based on this recursive formula.

A328880 If n = Product (p_j^k_j) then a(n) = Product (a(pi(p_j)) + 1), where pi = A000720, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 6, 3, 2, 3, 8, 5, 6, 7, 6, 12, 2, 4, 6, 3, 8, 9, 10, 4, 6, 4, 14, 3, 6, 9, 24, 6, 2, 15, 8, 12, 6, 7, 6, 21, 8, 8, 18, 7, 10, 12, 8, 13, 6, 3, 8, 12, 14, 3, 6, 20, 6, 9, 18, 5, 24, 7, 12, 9, 2, 28, 30, 4, 8, 12, 24, 9, 6, 10, 14, 12, 6, 15, 42, 11, 8

Offset: 1

Ilya Gutkovskiy, Oct 29 2019

			a(36) = 6 because 36 = 2^2 * 3^2 = prime(1)^2 * prime(2)^2 and (a(1) + 1) * (a(2) + 1) = (1 + 1) * (2 + 1) = 6.

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A048250, A156061, A184160, A225395, A282446, A328879.

a[1] = 1; a[n_] := Times @@ (a[PrimePi[#[[1]]]] + 1 & /@ FactorInteger[n]); Table[a[n], {n, 1, 80}]

a(n)={my(f=factor(n)[,1]); prod(i=1, #f, 1 + a(primepi(f[i])))} \\ Andrew Howroyd, Oct 29 2019

a(n) = a(prime(n)) - 1.

A184162 Number of chains in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 3, 7, 5, 15, 9, 11, 7, 13, 17, 31, 11, 19, 13, 21, 9, 23, 15, 15, 19, 17, 33, 27, 13, 29, 21, 19, 15, 35, 23, 63, 11, 37, 25, 25, 17, 23, 17, 25, 21, 39, 19, 27, 35, 27, 29, 43, 15, 21, 31, 29, 23, 19, 21, 45, 17, 21, 37, 47, 25, 31, 65, 23, 13, 33, 39, 31, 27, 33, 27, 39, 19, 35, 25, 35, 19, 41, 27, 67, 23

Offset: 1

Emeric Deutsch, Oct 19 2011

nonn

The vertices of a rooted tree can be regarded as a partially ordered set, where u<=v holds for two vertices u and v if and only if u lies on the unique path between v and the root. A chain is a nonempty set of pairwise comparable vertices.

			a(5) = 15 because the rooted tree with Matula-Goebel number 5 is a path ABCD on 4 vertices and any nonempty subset of {A,B,C,D} is a chain.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
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.
Index entries for sequences related to Matula-Goebel numbers

Cf. A109082 (height), A196056 (vertices at levels).
Cf. A184160 (antichains).
Cf. A049084, A020639.

import Data.List (genericIndex)
a184162 n = genericIndex a184162_list (n - 1)
a184162_list = 1 : g 2 where
   g x = y : g (x + 1) where
     y = if t > 0 then 2 * a184162 t + 1 else a184162 r + a184162 s - 1
         where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013

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 1 elif bigomega(n) = 1 then 1+2*a(pi(n)) else a(r(n))+a(s(n))-1 end if end proc: seq(a(n), n = 1 .. 80);

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
a[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + 2*a[PrimePi[n]], True, a[r[n]] + a[s[n]] - 1];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 24 2024, after Maple code *)

a(n) = my(f=factor(n)); [self()(primepi(p)) |p<-f[,1]] * f[,2]*2 + 1; \\ Kevin Ryde, Aug 25 2021

a(1)=1; if n=prime(t), then a(n)=1+2a(t); if n=r*s (r,s,>=2), then a(n)=a(r)+a(s)-1. The Maple program is based on this recursive formula.

a(n) = 1 + Sum_{k=1..A109082(n)} A196056(n,k)*2^k. - Kevin Ryde, Aug 25 2021

A198338 Irregular triangle read by rows: row n is the sequence of Matula numbers of the rooted subtrees of the rooted tree with Matula-Goebel number n. A root subtree of a rooted tree T is a subtree of T containing the root.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 3, 5, 1, 2, 2, 3, 4, 6, 1, 2, 3, 3, 7, 1, 2, 2, 2, 4, 4, 4, 8, 1, 2, 2, 3, 3, 4, 6, 6, 9, 1, 2, 2, 3, 4, 5, 6, 10, 1, 2, 3, 5, 11, 1, 2, 2, 2, 3, 4, 4, 4, 6, 6, 8, 12, 1, 2, 3, 3, 4, 6, 6, 7, 15, 1, 2, 2, 3, 3, 4, 5, 6, 6

Offset: 1

Emeric Deutsch, Dec 04 2011

Number of entries in row n is A184160(n). Row n>=2 can be easily identified: it starts with the entry following the first occurrence of n-1 and it ends with the first occurrence of n.

			Row 4 is [1,2,2,4] because the rooted tree with Matula-Goebel number 4 is V and its root subtrees are *, |, |, and V.
Triangle starts:
1;
1,2;
1,2,3;
1,2,2,4;
1,2,3,5;
1,2,2,3,4,6;

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 Review, 10, 1968, 273.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288

Cf. A198339.

with(numtheory): MRST := 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 [1] elif bigomega(n) = 1 then sort([1, seq(ithprime(mrst[pi(n)][i]), i = 1 .. nops(mrst[pi(n)]))]) else sort([seq(seq(mrst[r(n)][i]*mrst[s(n)][j], i = 1 .. nops(mrst[r(n)])), j = 1 .. nops(mrst[s(n)]))]) end if end proc: for n to 1000 do mrst[n] := MRST(n) end do;

Row 1 is [1]; if n = p(t) (= the t-th prime), then row n is [1, p(a), p(b), ... ], where [a,b,...] is row t; if n=rs (r,s >=2), then row n consists of the numbers r[i]*s[j], where [r[1], r[2],...] is row r and [s[1], s[2], ...] is row s. The Maple program, based on this recursive procedure, yields row n (<=1000; adjustable) with the command MRST(n).

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A206491 Irregular triangle read by rows: T(n,k) is the number of root subtrees with k nodes in the rooted tree having Matula-Goebel number n.

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A184161 Number of subtrees in the rooted tree with Matula-Goebel number n.

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A328880 If n = Product (p_j^k_j) then a(n) = Product (a(pi(p_j)) + 1), where pi = A000720, with a(1) = 1.

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A184162 Number of chains in the rooted tree with Matula-Goebel number n.

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A198338 Irregular triangle read by rows: row n is the sequence of Matula numbers of the rooted subtrees of the rooted tree with Matula-Goebel number n. A root subtree of a rooted tree T is a subtree of T containing the root.

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