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A196050 Number of edges in the rooted tree with Matula-Goebel number n. 84
0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 5, 5, 5, 6, 5, 6, 5, 5, 6, 5, 5, 6, 5, 6, 6, 5, 5, 6, 6, 5, 6, 5, 6, 7, 6, 6, 6, 6, 7, 6, 6, 5, 7, 7, 6, 6, 6, 5, 7, 6, 6, 7, 6, 7, 7, 5, 6, 7, 7, 6, 7, 6, 6, 8, 6, 7, 7, 6, 7, 8, 6, 6, 7, 7, 6, 7, 7, 6, 8, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 6, 7, 7, 7, 8, 6, 6, 8, 6, 8 (list; graph; refs; listen; history; text; internal format)
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.
a(n) is, for n >= 2, the number of prime function prime(.) = A000040(.) operations in the complete reduction of n. See the W. Lang link with a list of the reductions for n = 2..100, where a curly bracket notation {.} is used for prime(.). - Wolfdieter Lang, Apr 03 2018
From Gus Wiseman, Mar 23 2019: (Start)
Every positive integer has a unique factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
11 = q(1) q(2) q(3) q(5)
50 = q(1)^3 q(2)^2 q(3)^2
360 = q(1)^6 q(2)^3 q(3)
In this factorization, a(n) is the number of factors counted with multiplicity. For example, a(11) = 4, a(50) = 7, a(360) = 10.
(End)
From Antti Karttunen, Oct 23 2023: (Start)
Totally additive with a(prime(n)) = 1 + a(n).
Number of iterations of A366385 (or equally, of A366387) needed to reach 1.
(End)
LINKS
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Göbel, 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)=1 + a(t); if n = r*s (r,s>=2), then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.
a(n) = A061775(n) - 1.
a(n) = A109129(n) + A366388(n) = A109082(n) + A358729(n). - Antti Karttunen, Oct 23 2023
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 the star tree 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 bigomega(n) = 1 then 1+a(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);
MATHEMATICA
a[1] = 0; a[n_?PrimeQ] := a[n] = 1 + a[PrimePi[n]]; a[n_] := Total[#[[2]] * a[#[[1]] ]& /@ FactorInteger[n]];
Array[a, 110] (* Jean-François Alcover, Nov 16 2017 *)
difac[n_]:=If[n==1, {}, With[{i=PrimePi[FactorInteger[n][[1, 1]]]}, Sort[Prepend[difac[n*i/Prime[i]], i]]]];
Table[Length[difac[n]], {n, 100}] (* Gus Wiseman, Mar 23 2019 *)
PROG
(Haskell)
import Data.List (genericIndex)
a196050 n = genericIndex a196050_list (n - 1)
a196050_list = 0 : g 2 where
g x = y : g (x + 1) where
y = if t > 0 then a196050 t + 1 else a196050 r + a196050 s
where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013
(PARI) a(n) = my(f=factor(n)); [self()(primepi(p))+1 |p<-f[, 1]]*f[, 2]; \\ Kevin Ryde, May 28 2021
(Python)
from functools import lru_cache
from sympy import isprime, primepi, factorint
@lru_cache(maxsize=None)
def A196050(n):
if n == 1 : return 0
if isprime(n): return 1+A196050(primepi(n))
return sum(e*A196050(p) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022
CROSSREFS
One less than A061775.
Sequence in context: A265370 A356895 A238407 * A334097 A122027 A359121
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 27 2011
STATUS
approved

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Last modified June 24 04:41 EDT 2024. Contains 373661 sequences. (Running on oeis4.)