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

%I A325612 #12 Feb 25 2025 01:55:20
%S A325612 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 A325612 11,12,15,16,10,19,19,15,18,16,16,18,10,18,18,17,15,21,15,18,24,23,19,
%U A325612 23,25,25,18,26,25,28,21,21,25,23,21,29,28,31,21,24,23
%N A325612 Width (number of leaves) of the rooted tree with Matula-Goebel number 2^n - 1.
%C A325612 Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
%C A325612    11 = q(1) q(2) q(3) q(5)
%C A325612    50 = q(1)^3 q(2)^2 q(3)^2
%C A325612   360 = q(1)^6 q(2)^3 q(3)
%C A325612 For n > 1, a(n) is the multiplicity of q(1) = 2 in the q-factorization of 2^n - 1.
%H A325612 Keith Briggs, <a href="http://keithbriggs.info/matula.html">Matula numbers and rooted trees</a>.
%e A325612 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 A325612 mglv[n_]:=If[n==1,1,Total[Cases[FactorInteger[n],{p_,k_}:>mglv[PrimePi[p]]*k]]];
%t A325612 Table[mglv[2^n-1],{n,30}]
%Y A325612 Cf. A001222, A001221, A056239, A112798.
%Y A325612 Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.
%Y A325612 Mersenne numbers: A046051, A046800, A059305, A325610, A325611, A325625.
%K A325612 nonn
%O A325612 1,3
%A A325612 _Gus Wiseman_, May 12 2019
%E A325612 More terms from _Jinyuan Wang_, Feb 25 2025