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A342507 Number of internal nodes in rooted tree with Matula-Goebel number n.

%I A342507 #20 Dec 01 2022 17:44:48
%S A342507 0,1,2,1,3,2,2,1,3,3,4,2,3,2,4,1,3,3,2,3,3,4,4,2,5,3,4,2,4,4,5,1,5,3,
%T A342507 4,3,3,2,4,3,4,3,3,4,5,4,5,2,3,5,4,3,2,4,6,2,3,4,4,4,4,5,4,1,5,5,3,3,
%U A342507 5,4,4,3,4,3,6,2,5,4,5,3,5,4,5,3,5,3,5,4,3,5,4,4,6,5,4,2,6,3,6,5
%N A342507 Number of internal nodes in rooted tree with Matula-Goebel number n.
%C A342507 The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product_{T_i} prime(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it. (Cf. A061773 for illustration.)
%H A342507 François Marques, <a href="/A342507/b342507.txt">Table of n, a(n) for n = 1..10000</a>
%H A342507 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A342507 a(1)=0 and a(n) = A061775(n) - A109129(n) for n > 1.
%e A342507 a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
%e A342507 a(2^m) = 1 because the rooted tree with Matula-Goebel number 2^m is the star tree with m edges.
%t A342507 MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A342507 Table[Count[MGTree[n],_[__],{0,Infinity}],{n,100}] (* _Gus Wiseman_, Nov 28 2022 *)
%o A342507 (PARI) A342507(n) = if( n==1, 0, my(f=factor(n)); 1+sum(k=1,matsize(f)[1],A342507(primepi(f[k,1]))*f[k,2]));
%Y A342507 Other statistics are: A061775 (nodes), A109082 (edge-height), A109129 (leaves), A196050 (edges), A358552 (node-height).
%Y A342507 An ordered version is A358553.
%Y A342507 Positions of first appearances are A358554.
%Y A342507 A000081 counts rooted trees, ordered A000108.
%Y A342507 A358575 counts rooted trees by nodes and internals.
%Y A342507 Cf. A000040, A000720, A001222, A007097, A056239, A112798.
%Y A342507 Cf. A034781, A055277, A206487, A358576, A358578, A358592.
%K A342507 nonn
%O A342507 1,3
%A A342507 _François Marques_, Mar 14 2021