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 A196047 Path length of the rooted tree with Matula-Goebel number n. 8
 0, 1, 3, 2, 6, 4, 5, 3, 6, 7, 10, 5, 8, 6, 9, 4, 9, 7, 7, 8, 8, 11, 11, 6, 12, 9, 9, 7, 12, 10, 15, 5, 13, 10, 11, 8, 10, 8, 11, 9, 13, 9, 11, 12, 12, 12, 15, 7, 10, 13, 12, 10, 9, 10, 16, 8, 10, 13, 14, 11, 13, 16, 11, 6, 14, 14, 12, 11, 14, 12, 14, 9, 14, 11, 15, 9, 15, 12, 17, 10, 12, 14, 17, 10, 15, 12, 15, 13, 12, 13, 13, 13, 18, 16, 13, 8, 19, 11, 16, 14 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The path length of a rooted tree is defined as the sum of distances of all nodes to the root of the tree. 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 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. 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; if n=p(t) (= the t-th prime) then a(n)=a(t)+N(t), where N(t) is the number of nodes of 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(n) = A196048(n) + A343006(n). - François Marques, Apr 02 2021 EXAMPLE a(7) = 5 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (0+1+2+2 = 5). a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges. MAPLE with(numtheory): a := proc (n) local r, s, N: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: N := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then 1+N(pi(n)) else N(r(n))+N(s(n))-1 end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n))+N(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 100); MATHEMATICA a[m_] := Module[{r, s, Nn}, r[n_] := FactorInteger[n][[1, 1]]; s[n_] := n/r[n]; Nn[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1+Nn[PrimePi[n]], True, Nn[r[n]]+Nn[s[n]]-1]; Which[m == 1, 0, PrimeOmega[m] == 1, a[PrimePi[m]]+Nn[PrimePi[m]], True, a[r[m]]+a[s[m]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 03 2023, after Maple code *) PROG (Haskell) import Data.List (genericIndex) a196047 n = genericIndex a196047_list (n - 1) a196047_list = 0 : g 2 where g x = y : g (x + 1) where y = if t > 0 then a196047 t + a061775 t else a196047 r + a196047 s where t = a049084 x; r = a020639 x; s = x `div` r -- Reinhard Zumkeller, Sep 03 2013 (PARI) NPl(n) = { if(n==1, return([1, 0]), my(f=factor(n)~, v=Mat(vector(#f, k, NPl(primepi(f[1, k]))~)) ); return( [ 1+sum(k=1, #f, v[1, k]*f[2, k]) , sum(k=1, #f, (v[1, k]+v[2, k])*f[2, k]) ] ) ) }; A196047(n) = NPl(n)[2]; \\ François Marques, Apr 02 2021 CROSSREFS Cf. A049084, A020639. Cf. A196048, A343006. Sequence in context: A340873 A033940 A286367 * A106409 A115510 A230598 Adjacent sequences: A196044 A196045 A196046 * A196048 A196049 A196050 KEYWORD nonn AUTHOR Emeric Deutsch, Sep 27 2011 STATUS approved

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Last modified September 21 12:00 EDT 2023. Contains 365501 sequences. (Running on oeis4.)