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A196057
Number of sibling pairs in the rooted tree with Matula-Goebel number n.
1
0, 0, 0, 1, 0, 1, 1, 3, 1, 1, 0, 3, 1, 2, 1, 6, 1, 3, 3, 3, 2, 1, 1, 6, 1, 2, 3, 4, 1, 3, 0, 10, 1, 2, 2, 6, 3, 4, 2, 6, 1, 4, 2, 3, 3, 2, 1, 10, 3, 3, 2, 4, 6, 6, 1, 7, 4, 2, 1, 6, 3, 1, 4, 15, 2, 3, 3, 4, 2, 4, 3, 10, 2, 4, 3, 6, 2, 4, 1, 10, 6, 2, 1, 7, 2, 3, 2, 6, 6, 6, 3, 4, 1, 2, 4, 15, 1, 5, 3, 6, 2, 4, 3, 7, 4, 7, 4, 10, 1, 3
OFFSET
1,8
COMMENTS
A sibling pair is an unordered pair of nodes having the same parent.
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; if n=prime(t) (=the t-th prime), then a(n)=a(t); if n=r*s (r,s,>=2), then a(n) = a(r) + a(s) + G(r)*G(s), where G(m) is the number of prime divisors of m, counted with multiplicities. The Maple program is based on this recursive formula.
EXAMPLE
a(7)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = binomial(m,2) because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
MAPLE
with(numtheory): a := proc (n) local u, v: u := proc (n) options operator, arrow: op(1, factorset(n)) end proc: v := proc (n) options operator, arrow: n/u(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n)) else a(u(n))+a(v(n))+bigomega(u(n))*bigomega(v(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, PrimeOmega[n] == 1, a[PrimePi[n]], True, a[r[n]] + a[s[n]] + PrimeOmega[r[n]]*PrimeOmega[s[n]]];
Table[a[n], {n, 1, 110}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)
PROG
(Haskell)
import Data.List (genericIndex)
a196057 n = genericIndex a196057_list (n - 1)
a196057_list = 0 : g 2 where
g x = y : g (x + 1) where
y | t > 0 = a196057 t
| otherwise = a196057 r + a196057 s + a001222 r * a001222 s
where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 30 2011
STATUS
approved