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 A196048 External path length of the rooted tree with Matula-Goebel number n. 6
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The external path length of a rooted tree is defined as the sum of the distances of all leaves to the root. 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. LINKS François Marques, 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. Goebel, 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. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273. Index entries for sequences related to Matula-Goebel numbers FORMULA 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. a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges. EXAMPLE a(7)=4 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (2+2=4). MAPLE 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); MATHEMATICA a[m_] := Module[{r, s, LV}, r[n_] := FactorInteger[n][[1, 1]]; s[n_] := n/r[n]; LV [n_] := Which[ n == 1, 0, n == 2, 1, PrimeOmega[n] == 1, LV[PrimePi[n]], True, LV[r[n]] + LV[s[n]]]; Which[ m == 1, 0, m == 2, 1, PrimeOmega[m] == 1, a[PrimePi[m]] + LV[PrimePi[m]], True, a[r[m]] + a[s[m]]]]; Table[a[n], {n, 1, 110}] (* Jean-François Alcover, May 04 2023, after Maple code *) PROG (Haskell) import Data.List (genericIndex) a196048 n = genericIndex a196048_list (n - 1) a196048_list = 0 : 1 : g 3 where g x = y : g (x + 1) where y = if t > 0 then a196048 t + a109129 t else a196048 r + a196048 s where t = a049084 x; r = a020639 x; s = x `div` r -- Reinhard Zumkeller, Sep 03 2013 (PARI) LEpl(n) = { if(n==1, return([1, 0]), my(f=factor(n)~, l, e, le); foreach(f, p, le=LEpl(primepi(p)); l+=le*p; e+=(le+le)*p; ); return([l, e]) ) }; A196048(n) = LEpl(n); \\ François Marques, Mar 14 2021 CROSSREFS Cf. A109129, A196047. Cf. A049084, A020639. Sequence in context: A157790 A070241 A066412 * A176075 A256555 A117119 Adjacent sequences: A196045 A196046 A196047 * A196049 A196050 A196051 KEYWORD nonn AUTHOR Emeric Deutsch, Sep 27 2011 EXTENSIONS Offset fixed by Reinhard Zumkeller, Sep 03 2013 STATUS approved

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Last modified December 2 07:49 EST 2023. Contains 367510 sequences. (Running on oeis4.)