OFFSET
1,3
COMMENTS
The Matula-Goebel number of a rooted tree is 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.
REFERENCES
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 Review, 10, 1968, 273.
LINKS
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
FORMULA
a(1)=0; if n is the t-th prime, then a(n) = 1 + a(t); if n is composite, n=rs, then a(n) = min(a(r),a(s)). The Maple program is based on this recursive rule.
EXAMPLE
a(7)=2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (all leaves are at level 2).
MAPLE
with (numtheory):
a:= proc(n) option remember;
if n=1 then 0
elif isprime(n)=true then 1 +a(pi(n))
else min (seq (a(i), i=factorset(n)))
fi
end:
seq (a(n), n=1..200);
MATHEMATICA
a[n_] := a[n] = Which[
n == 1, 0,
PrimeQ[n], 1 + a[PrimePi[n]],
True, Min[a /@ FactorInteger[n][[All, 1]]]];
Table[a[n], {n, 1, 200}] (* Jean-François Alcover, Jun 24 2024, after Maple code *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 22 2011
STATUS
approved