OFFSET
1,2
COMMENTS
Number of entries in row n is A235122(n).
LINKS
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
Emerci 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.
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
MATHEMATICA
f[m_] := Prime[m/#]*PrimePi[#]& /@ FactorInteger[m][[All, 1]];
M[m_] := Module[{A, B}, A = {m}; While[True, B = A; A = Union[Map[f, A] // Flatten, A] // DeleteCases[#, 0]&; If[B == A, Return[A]]]];
Table[M[j], {j, 1, 25}] // Flatten (* Jean-François Alcover, Sep 14 2024, after W. Edwin Clark *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jan 18 2014
STATUS
approved