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A325612 Width (number of leaves) of the rooted tree with Matula-Goebel number 2^n - 1. 4


%S 1,1,2,2,1,4,1,4,5,3,6,7,4,5,7,6,7,11,7,7,9,10,7,13,7,11,9,11,11,13,

%T 11,12,15,16,10,19,19,15,18,16,16,18,10,18,18,17,15,21,15,18,24,23,19,

%U 23,25,25,18,26,25,28

%N Width (number of leaves) of the rooted tree with Matula-Goebel number 2^n - 1.

%C Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:

%C 11 = q(1) q(2) q(3) q(5)

%C 50 = q(1)^3 q(2)^2 q(3)^2

%C 360 = q(1)^6 q(2)^3 q(3)

%C For n > 1, a(n) is the multiplicity of q(1) = 2 in the q-factorization of 2^n - 1.

%H Keith Briggs, <a href="http://keithbriggs.info/matula.html">Matula numbers and rooted trees.</a>

%e The rooted tree with Matula-Goebel number 2047 = 2^11 - 1 is (((o)(o))(ooo(o))), which has 6 leaves (o's), so a(11) = 6.

%t mglv[n_]:=If[n==1,1,Total[Cases[FactorInteger[n],{p_,k_}:>mglv[PrimePi[p]]*k]]];

%t Table[mglv[2^n-1],{n,30}]

%Y Cf. A001222, A001221, A056239, A112798.

%Y Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.

%Y Mersenne numbers: A046051, A046800, A059305, A325610, A325611, A325625.

%K nonn,more

%O 1,3

%A _Gus Wiseman_, May 12 2019

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Last modified December 8 04:31 EST 2019. Contains 329850 sequences. (Running on oeis4.)