

A257540


Irregular triangle read by rows: row n (n>=2) contains the degrees of the level 1 vertices of the rooted tree having MatulaGoebel number n; row 1: 0.


0



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

1,3


COMMENTS

The Matula (or MatulaGoebel) 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 MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.
Number of entries in row n is the number of prime divisors of n counted with multiplicity.
Sum of entries in row n = A196052(n).


LINKS



FORMULA

Denoting by DL(n) the multiset of the degrees of the level 1 vertices of the rooted tree with Matula number n, we have DL(1)=[0], DL[2]=[1], DL(ith prime) = [1+bigomega(i)], DL(rs) = DL(r) union DL(s), where bigomega(i) is the number of prime divisors of i, counted with multiplicity (A001222) and "union" is "multiset union". The Maple program is based on these recurrence equations.


EXAMPLE

Row 8 is 1,1,1. Indeed, the rooted tree with Matula number 8 is the star tree \/; vertices at level 1 have degrees 1,1,1.
Triangle starts:
0;
1;
2;
1,1;
2;
1,2;
3;
1,1,1;


MAPLE

with(numtheory): DL := proc (n) if n = 2 then [1] elif bigomega(n) = 1 then [1+bigomega(pi(n))] else [op(DL(op(1, factorset(n)))), op(DL(n/op(1, factorset(n))))] end if end proc: with(numtheory): DL := proc (n) if n = 1 then [0] elif n = 2 then [1] elif bigomega(n) = 1 then [1+bigomega(pi(n))] else [op(DL(op(1, factorset(n)))), op(DL(n/op(1, factorset(n))))] end if end proc: seq(op(DL(n)), n = 1 .. 100);


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



