

A109082


Depth of rooted tree having MatulaGoebel 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
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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 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=0; if n is the tth 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 MatulaGoebel number 7 is the 3edge 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}] (* JeanFrançois 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



