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Number of nodes in the rooted tree with Matula-Goebel number 2^n - 1.
7

%I #5 May 13 2019 01:10:01

%S 1,3,4,6,6,8,7,10,10,12,12,15,12,14,16,18,14,20,16,23,20,22,22,25,25,

%T 24,23,29,26,30,27,31,33,28,32,38,36,31,36,40,37,38,33,43,44,42,39,48,

%U 39,49,45,48,43,49,49,53,47,54,47,61

%N Number of nodes in 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 Then a(n) is one plus the number of factors (counted with multiplicity) in the q-factorization of 2^n - 1.

%e The rooted tree with Matula-Goebel number 2047 = 2^11 - 1 is (((o)(o))(ooo(o))), which has 12 nodes (o's plus brackets), so a(11) = 12.

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

%t Table[mgwt[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, A325612, A325625.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 12 2019