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 A109082 Depth of rooted tree having Matula-Goebel number n. 34
 0, 1, 2, 1, 3, 2, 2, 1, 2, 3, 4, 2, 3, 2, 3, 1, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 2, 2, 4, 3, 5, 1, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 4, 3, 3, 4, 2, 2, 3, 3, 3, 2, 2, 4, 2, 2, 4, 4, 3, 3, 5, 2, 1, 3, 4, 3, 3, 3, 3, 4, 2, 3, 3, 3, 2, 4, 3, 5, 3, 2, 4, 4, 2, 3, 3, 4, 4, 3, 3, 3, 3, 5, 4, 3, 2, 4, 2, 4, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Another term for depth is height. Starting with n, a(n) is the number of times one must take the product of prime indices (A003963) to reach 1. - Gus Wiseman, Mar 27 2019 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..5381 E. 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. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273. FORMULA a(1)=0; if n is the t-th prime, then a(n) = 1 + a(t); if n is composite, n=ts, then a(n) = max(a(t),a(s)). The Maple program is based on this. a(A007097(n)) = n. EXAMPLE a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the 3-edge rooted tree Y of height 2. MAPLE with(numtheory): a := proc(n) option remember; if n = 1 then 0 elif isprime(n) then 1+a(pi(n)) else max((map (p->a(p), factorset(n)))[]) end if end proc: seq(a(n), n = 1 .. 100); # Emeric Deutsch, Sep 16 2011 MATHEMATICA a [n_] := a[n] = If[n == 1, 0, If[PrimeQ[n], 1+a[PrimePi[n]], Max[Map[a, FactorInteger[n][[All, 1]]]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 06 2014, after Emeric Deutsch *) CROSSREFS A left inverse of A007097. Cf. A003963, A061775, A091233. Cf. A000081, A000720, A001222, A109129, A112798, A196050, A290822, A317713, A320325, A324927 (positions of 2), A324928 (positions of 3), A325032. Sequence in context: A096857 A303639 A090000 * A324923 A126303 A306467 Adjacent sequences:  A109079 A109080 A109081 * A109083 A109084 A109085 KEYWORD nonn AUTHOR Keith Briggs, Aug 17 2005 EXTENSIONS Edited by Emeric Deutsch, Sep 16 2011 STATUS approved

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Last modified August 15 10:51 EDT 2020. Contains 336492 sequences. (Running on oeis4.)