%I
%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 MatulaGoebel number 2^n  1.
%C Every positive integer has a unique qfactorization (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 qfactorization 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 MatulaGoebel 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^n1],{n,30}]
%Y Cf. A001222, A001221, A056239, A112798.
%Y MatulaGoebel 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
