

A257539


The smallest of the Matula numbers of the rooted trees that are isomorphic as trees to the rooted tree with Matula number n.


1



1, 2, 3, 3, 5, 5, 7, 7, 9, 9, 9, 12, 12, 12, 15, 16, 12, 18, 16, 20, 20, 15, 18, 24, 25, 18, 27, 28, 20, 30, 15, 32, 25, 20, 35, 36, 24, 24, 30, 40, 18, 42, 28, 35, 45, 27, 30, 48, 49, 50, 35, 42, 32, 54, 55, 56, 40, 30, 20, 60, 36, 25, 63, 64, 65, 65, 24, 49, 45, 70, 40
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OFFSET

1,2


COMMENTS

The Matula number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the Matula 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 numbers of the m branches of T.


REFERENCES

E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Applied Math., 160, 2012, 23142322.
F. Goebel, On a 11 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YN. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
J. L. Martin, M. Morin, J. D. Wagner, On distinguishing trees by their chromatic symmetric functions, J. Combin. Theory, A115, 2008, 237253.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.


LINKS

Table of n, a(n) for n=1..71.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(n) = first entry in row n of A235121. The Maple program is based on this.


EXAMPLE

a(4) = 3. Indeed the rooted tree corresponding to the Matula number 4 is \/ . It is isomorphic as a tree only to itself and to the path tree P(3), rooted at one of its endvertex; the Matula number of this tree is 3; min {3,4} = 3.
a(14)=12 because the Matula numbers of the rooted trees isomorphic as a tree to the rooted tree having Matula number 14 are: 12, 13, 14, and 17.


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: seq(M(j)[1], j = 1 .. 100);


CROSSREFS

Cf. A235121.
Sequence in context: A239906 A239907 A113637 * A159050 A210336 A280271
Adjacent sequences: A257536 A257537 A257538 * A257540 A257541 A257542


KEYWORD

nonn


AUTHOR

Emeric Deutsch, May 02 2015


STATUS

approved



