

A196050


Number of edges in the rooted tree with MatulaGoebel number n.


80



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
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OFFSET

1,3


COMMENTS

The MatulaGoebel number of a rooted tree is 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.
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)


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Göbel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
Wolfdieter Lang, Complete prime function reduction for n = 2..100.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=0; if n = prime(t) (the tth 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.


EXAMPLE

a(7) = 3 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y.
a(2^m) = m because the rooted tree with MatulaGoebel 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] (* JeanFrançois 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.
Cf. A000040, A000081, A000720, A003963, A007097, A020639, A049084, A109082, A109129, A317713.
Cf. A324850, A324922, A324923, A324924, A324925, A324931, A324935.
Sequence in context: A265370 A356895 A238407 * A334097 A122027 A359121
Adjacent sequences: A196047 A196048 A196049 * A196051 A196052 A196053


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Sep 27 2011


STATUS

approved



