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A196052 Sum of the degrees of the nodes at level 1 in the rooted tree with Matula-Goebel number n.

%I A196052 #31 Jun 26 2024 02:24:14
%S A196052 0,1,2,2,2,3,3,3,4,3,2,4,3,4,4,4,2,5,4,4,5,3,3,5,4,4,6,5,3,5,2,5,4,3,
%T A196052 5,6,4,5,5,5,2,6,3,4,6,4,3,6,6,5,4,5,5,7,4,6,6,4,2,6,4,3,7,6,5,5,2,4,
%U A196052 5,6,4,7,3,5,6,6,5,6,3,6,8,3,2,7,4,4,5,5,5,7,6,5,4,4,6,7,3,7,6,6,3,5,4,6,7,6,4,8,2,5
%N A196052 Sum of the degrees of the nodes at level 1 in the rooted tree with Matula-Goebel number n.
%C A196052 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.
%D A196052 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A196052 Reinhard Zumkeller, <a href="/A196052/b196052.txt">Table of n, a(n) for n = 1..10000</a>
%H A196052 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
%H A196052 F. Goebel, <a href="https://doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H A196052 I. Gutman and A. Ivic, <a href="https://doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A196052 I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H A196052 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A196052 a(1)=0; if n = prime(t) (the t-th prime), then a(n)=1+G(t), where G(t) is the number of prime divisors of t counted with multiplicities;  if n=r*s (r,s>=2), then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.
%e A196052 a(7)=3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
%e A196052 a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%p A196052 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+bigomega(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);
%t A196052 r[n_] := FactorInteger[n][[1, 1]];
%t A196052 s[n_] := n/r[n];
%t A196052 a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, 1 + PrimeOmega[PrimePi[n]], True, a[r[n]] + a[s[n]]];
%t A196052 Table[a[n], {n, 1, 110}] (* _Jean-François Alcover_, Jun 25 2024, after Maple code *)
%o A196052 (Haskell)
%o A196052 import Data.List (genericIndex)
%o A196052 a196052 n = genericIndex a196052_list (n - 1)
%o A196052 a196052_list = 0 : g 2 where
%o A196052    g x = y : g (x + 1) where
%o A196052      y = if t > 0 then a001222 t + 1 else a196052 r + a196052 s
%o A196052          where t = a049084 x; r = a020639 x; s = x `div` r
%o A196052 -- _Reinhard Zumkeller_, Sep 03 2013
%o A196052 (PARI) a(n) = my(m=factor(n)); [bigomega(primepi(p))+1 | p<-m[,1]] * m[,2]; \\ _Kevin Ryde_, Oct 16 2020
%Y A196052 Cf. A049084, A020639, A001222.
%K A196052 nonn
%O A196052 1,3
%A A196052 _Emeric Deutsch_, Sep 27 2011