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%I #64 Nov 28 2022 10:02:10
%S 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,
%T 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,
%U 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
%N Depth of rooted tree having Matula-Goebel number n.
%C Another term for depth is height.
%C 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
%H François Marques, <a href="/A109082/b109082.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5381 from Alois P. Heinz).
%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. W. 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; if n is the t-th prime, then a(n) = 1 + a(t); if n is composite, n=t*s, then a(n) = max(a(t),a(s)). The Maple program is based on this.
%F a(A007097(n)) = n.
%F a(n) = A358552(n) - 1. - _Gus Wiseman_, Nov 27 2022
%e a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the 3-edge rooted tree Y of height 2.
%p 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
%t 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_ *)
%o (PARI) a(n) = my(v=factor(n)[,1],d=0); while(#v,d++; v=fold(setunion, apply(p->factor(primepi(p))[,1]~, v))); d; \\ _Kevin Ryde_, Sep 21 2020
%o (Python)
%o from functools import lru_cache
%o from sympy import isprime, primepi, primefactors
%o @lru_cache(maxsize=None)
%o def A109082(n):
%o if n == 1 : return 0
%o if isprime(n): return 1+A109082(primepi(n))
%o return max(A109082(p) for p in primefactors(n)) # _Chai Wah Wu_, Mar 19 2022
%Y A left inverse of A007097.
%Y Cf. A003963, A061775, A091233.
%Y Cf. A000081, A000720, A001222, A109129, A112798, A196050, A290822, A317713, A320325, A324927 (positions of 2), A324928 (positions of 3), A325032.
%Y This statistic is counted by A034781, ordered A080936.
%Y The ordered version is A358379.
%Y For node-height instead of edge-height we have A358552.
%Y Cf. A000108, A055277, A056239, A342507, A358576.
%K nonn
%O 1,3
%A _Keith Briggs_, Aug 17 2005
%E Edited by _Emeric Deutsch_, Sep 16 2011
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