login
A196067
Number of pendant vertices in the rooted tree with Matula-Goebel number n.
0
0, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 2, 4, 3, 3, 4, 3, 3, 2, 3, 4, 2, 3, 3, 4, 3, 3, 2, 5, 2, 3, 3, 4, 4, 4, 3, 4, 3, 4, 4, 3, 3, 3, 3, 5, 4, 3, 3, 4, 5, 4, 2, 5, 4, 3, 3, 4, 4, 2, 4, 6, 3, 3, 4, 4, 3, 4, 4, 5, 4, 4, 3, 5, 3, 4, 3, 5, 4, 3, 3, 5, 3, 4, 3, 4, 5, 4, 4, 4, 2, 3, 4, 6, 3, 5, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 3, 3
OFFSET
1,2
COMMENTS
A pendant vertex in a tree is a vertex having degree 1.
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.
LINKS
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.
FORMULA
a(1)=0; a(2)=2; if n=prime(t) (=the t-th prime) and t is prime, then a(n)=a(t); if n=prime(t) (the t-th prime) and t is not prime, then a(n)=1+a(t); if n=rs (r,s,>=2), then a(n)=a(r)+a(s)-the number of primes in {r,s}. The Maple program is based on this recursive formula.
EXAMPLE
a(7)=3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
MAPLE
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 0 elif n = 2 then 2 elif bigomega(n) = 1 and bigomega(pi(n)) = 1 then a(pi(n)) elif bigomega(n) = 1 then a(pi(n))+1 elif bigomega(r(n)) = 1 and bigomega(s(n)) = 1 then a(r(n))+a(s(n))-2 elif min(bigomega(r(n)), bigomega(s(n))) = 1 then a(r(n))+a(s(n))-1 else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);
MATHEMATICA
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
a[n_] := Which[n == 1, 0, n == 2, 2, PrimeOmega[n] == 1 && PrimeOmega[ PrimePi[n]] == 1, a[PrimePi[n]], PrimeOmega[n] == 1, a[PrimePi[n]] + 1, PrimeOmega[r[n]] == 1 && PrimeOmega[s[n]] == 1, a[r[n]] + a[s[n]] - 2, Min[PrimeOmega[r[n]], PrimeOmega[s[n]]] == 1, a[r[n]] + a[s[n]] - 1, True, a[r[n]] + a[s[n]]];
Table[a[n], {n, 1, 110}] (* Jean-François Alcover, Aug 07 2024, after Emeric Deutsch's Maple code *)
CROSSREFS
Sequence in context: A029230 A280945 A349949 * A251141 A319696 A320111
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 03 2011
STATUS
approved