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A196048 External path length of the rooted tree with Matula-Goebel number n.

%I A196048 #34 May 04 2023 15:32:51
%S A196048 0,1,2,2,3,3,4,3,4,4,4,4,5,5,5,4,6,5,6,5,6,5,6,5,6,6,6,6,6,6,5,5,6,7,
%T A196048 7,6,7,7,7,6,7,7,8,6,7,7,7,6,8,7,8,7,8,7,7,7,8,7,8,7,8,6,8,6,8,7,9,8,
%U A196048 8,8,8,7,9,8,8,8,8,8,7,7,8,8,8,8,9,9,8,7,9,8,9,8,7,8,9,7,8,9,8,8,9,9,9,8,9,9,10,8,8,8
%N A196048 External path length of the rooted tree with Matula-Goebel number n.
%C A196048 The external path length of a rooted tree is defined as the sum of the distances of all leaves to the root.
%C A196048 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.
%H A196048 François Marques, <a href="/A196048/b196048.txt">Table of n, a(n) for n = 1..10000</a>
%H A196048 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 A196048 F. Goebel, <a href="http://dx.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 A196048 I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A196048 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 A196048 D. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H A196048 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A196048 a(1)=0; a(2)=1; if n=prime(t) (the t-th prime; t>1) then a(n)=a(t)+LV(t), where LV(t) is the number of leaves in the rooted tree with Matula number t; if n=rs (r,s>=2), then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.
%F A196048 a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%e A196048 a(7)=4 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (2+2=4).
%p A196048 with(numtheory): a := proc (n) local r, s, LV: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LV := proc (n) if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then LV(pi(n)) else LV(r(n))+LV(s(n)) end if end proc: if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n))+LV(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);
%t A196048 a[m_] := Module[{r, s, LV},
%t A196048    r[n_] := FactorInteger[n][[1, 1]];
%t A196048    s[n_] := n/r[n];
%t A196048    LV [n_] := Which[
%t A196048       n == 1, 0,
%t A196048       n == 2, 1,
%t A196048       PrimeOmega[n] == 1, LV[PrimePi[n]],
%t A196048       True, LV[r[n]] + LV[s[n]]];
%t A196048    Which[
%t A196048       m == 1, 0,
%t A196048       m == 2, 1,
%t A196048       PrimeOmega[m] == 1, a[PrimePi[m]] + LV[PrimePi[m]],
%t A196048       True, a[r[m]] + a[s[m]]]];
%t A196048 Table[a[n], {n, 1, 110}] (* _Jean-François Alcover_, May 04 2023, after Maple code *)
%o A196048 (Haskell)
%o A196048 import Data.List (genericIndex)
%o A196048 a196048 n = genericIndex a196048_list (n - 1)
%o A196048 a196048_list = 0 : 1 : g 3 where
%o A196048    g x = y : g (x + 1) where
%o A196048      y = if t > 0 then a196048 t + a109129 t else a196048 r + a196048 s
%o A196048          where t = a049084 x; r = a020639 x; s = x `div` r
%o A196048 -- _Reinhard Zumkeller_, Sep 03 2013
%o A196048 (PARI) LEpl(n) = { if(n==1, return([1,0]),
%o A196048     my(f=factor(n)~, l, e, le);
%o A196048       foreach(f,p,
%o A196048         le=LEpl(primepi(p[1]));
%o A196048         l+=le[1]*p[2];
%o A196048         e+=(le[1]+le[2])*p[2];
%o A196048       );
%o A196048     return([l,e]) )
%o A196048   };
%o A196048 A196048(n) = LEpl(n)[2]; \\ _François Marques_, Mar 14 2021
%Y A196048 Cf. A109129, A196047.
%Y A196048 Cf. A049084, A020639.
%K A196048 nonn
%O A196048 1,3
%A A196048 _Emeric Deutsch_, Sep 27 2011
%E A196048 Offset fixed by _Reinhard Zumkeller_, Sep 03 2013