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A109082
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Depth of rooted tree having Matula-Goebel number n.
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57
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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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.
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EXAMPLE
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a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the 3-edge rooted tree Y of height 2.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
(Python)
from functools import lru_cache
from sympy import isprime, primepi, primefactors
@lru_cache(maxsize=None)
if n == 1 : return 0
if isprime(n): return 1+A109082(primepi(n))
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CROSSREFS
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Cf. A000081, A000720, A001222, A109129, A112798, A196050, A290822, A317713, A320325, A324927 (positions of 2), A324928 (positions of 3), A325032.
For node-height instead of edge-height we have A358552.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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