A196048 - OEIS

%I #34 May 04 2023 15:32:51

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

%C The external path length of a rooted tree is defined as the sum of the distances of all leaves to the root.

%C 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 François Marques, <a href="/A196048/b196048.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 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 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 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 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F 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 a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

%e a(7)=4 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (2+2=4).

%p 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 a[m_] := Module[{r, s, LV},

%t    r[n_] := FactorInteger[n][[1, 1]];

%t    s[n_] := n/r[n];

%t    LV [n_] := Which[

%t       n == 1, 0,

%t       n == 2, 1,

%t       PrimeOmega[n] == 1, LV[PrimePi[n]],

%t       True, LV[r[n]] + LV[s[n]]];

%t    Which[

%t       m == 1, 0,

%t       m == 2, 1,

%t       PrimeOmega[m] == 1, a[PrimePi[m]] + LV[PrimePi[m]],

%t       True, a[r[m]] + a[s[m]]]];

%t Table[a[n], {n, 1, 110}] (* _Jean-François Alcover_, May 04 2023, after Maple code *)

%o (Haskell)

%o import Data.List (genericIndex)

%o a196048 n = genericIndex a196048_list (n - 1)

%o a196048_list = 0 : 1 : g 3 where

%o    g x = y : g (x + 1) where

%o      y = if t > 0 then a196048 t + a109129 t else a196048 r + a196048 s

%o          where t = a049084 x; r = a020639 x; s = x `div` r

%o -- _Reinhard Zumkeller_, Sep 03 2013

%o (PARI) LEpl(n) = { if(n==1, return([1,0]),

%o     my(f=factor(n)~, l, e, le);

%o       foreach(f,p,

%o         le=LEpl(primepi(p[1]));

%o         l+=le[1]*p[2];

%o         e+=(le[1]+le[2])*p[2];

%o       );

%o     return([l,e]) )

%o   };

%o A196048(n) = LEpl(n)[2]; \\ _François Marques_, Mar 14 2021

%Y Cf. A109129, A196047.

%Y Cf. A049084, A020639.

%K nonn

%O 1,3

%A _Emeric Deutsch_, Sep 27 2011

%E Offset fixed by _Reinhard Zumkeller_, Sep 03 2013