A235121 Irregular triangle read by rows: row n contains in increasing order the Matula numbers of the rooted trees that are isomorphic as trees to the rooted tree with Matula number n (n>=1).

1, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 7, 8, 9, 10, 11, 9, 10, 11, 9, 10, 11, 12, 13, 14, 17, 12, 13, 14, 17, 12, 13, 14, 17, 15, 22, 31, 16, 19, 12, 13, 14, 17, 18, 23, 26, 41, 16, 19, 20, 21, 29, 34, 59, 20, 21, 29, 34, 59, 15, 22, 31, 18, 23, 26, 41, 24, 37, 38, 67, 25, 33, 62, 127

Offset

1,2

Comments

Number of entries in row n is A235122(n).

Links

Table of n, a(n) for n=1..75.

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.

D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.

Index entries for sequences related to Matula-Goebel numbers

Formula

In order to construct the set A of numbers corresponding to row n, we start with A = {n} and we keep adjoining to A the numbers (x/p)' p", where x is an element of A, p is a prime factor of x, r' denotes the r-th prime and r" denotes the order of the prime r (i.e. r = r"-th prime). We do this until A becomes closed under the described operation. The Maple program (due to Edwin Clark) is based on this construction.

Example

Row 11 is 9,10,11. Indeed the rooted tree with Matula number 11 is the path tree P[5] = ABCDE rooted at A; if rooted at B or D, then the Matula number is 10 and if rooted at C, then the Matula number is 9.
The triangle starts:
1;
2;
3,4;
3,4;
5,6;
5,6;
7,8;

Maple

with(numtheory): f := proc (m) local x, p, S: S := NULL: x := factorset(m): for p in x do S := S, ithprime(m/p)*pi(p) end do: S end proc: M := proc (m) local A, B: A := {m}: do B := A: A := `union`(map(f, A), A): if B = A then return A end if end do end proc: for j to 20 do M(j) end do; # yields sequence in triangular form; from W. Edwin Clark

Crossrefs

Cf. A235122.

Cf. A257539 (the first column).

Sequence in context: A308937 A123066 A330239 * A270652 A326693 A280242

Adjacent sequences: A235118 A235119 A235120 * A235122 A235123 A235124

Keyword

nonn,tabf

Author

Emeric Deutsch, Jan 18 2014

Status

approved